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a modern review 






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Physics for Teachers: 
A Modern Review 



Physics for Teachers: 
A Modern Review 



Robert L. Weber 

Associate Professor of Physics 
The Pennsylvania State University 



\ vV' 



McGraw-Hill Book Company 

New York Son Francisco Toronto London 



to Marion, Robert, Karen, 

Meredith, and Ruth 

who were patient 



-» 



PHYSICS FOR 



WIQAN UBRARfES 
WITH DRAWN FOR 




EW 



Copyright © 1964 by McGraw-Hill, Inc. All Rights Reserved. Printed in 
the United States of America, This book, or parts thereof, may not be 
reproduced in any form without permission of the publishers. 

Library of Congress Catalog Card Number 63-19320 
68806 



Preface 



The reader of this boob is assumed to be interested in space 
physics and atomic physics and to have had a full-year course in 
general physics at the college level. Capitalizing on his interest 
in rockets and satellites. Part 1 presents enough of the principles 
of mechanics and electricity to serve as a good basis for under- 
standing nineteenth-century classical physics. In Part 2, with 
the atom as the central theme, the theories of relativity and 
quantum physics which have characterized twentieth-century 
modern physics are developed. In both parts important topics 
such as wave properties and relativity, which are likely to be 
less familiar to the reader, are developed in greater detail. Some 
calculus notation is used,- but where feasible (e.g., in Sec. 4.4), 
noncalculus explanations are used, and several derivations in- 
volving integration are subordinated in an Appendix. An aim of 
this book is to encourage the reader, whether a student or a 
mature teacher, to appreciate the relatedness of the various 
fields of science and to be willing to venture into new areas with 
the ability he has gained from intensive study of a few selected 
areas. 

In the planning of this book I am indebted to the interest of 
students and colleagues in several science institutes sponsored by 
the National Science Foundation. For six years I served as 
vii 



Preface 



associate director, director, and teacher of the physics part of 
programs at The Pennsylvania State University; Colorado State 
University, and Yale University. The present text evolved from 
the study guide used in the last-mentioned program. I express 
indebtedness to that scholarly textbook "Physics for Students 
of Science and Engineering," by David Halliday and Robert 
Resniek, for the manner of presentation used in the first part of 
Chap. 13. 



Contents 



Robert L. Weber 



Preface vll 



Part 1 Looking Out: Rockets, Satellites, Space Travel 1 






What's Up? 3 

Environment of Space 9 

Rocket Propulsion 20 

Escape from Earth 40 

Satellites 50 

Motion of Bodies in Space 60 

Travel to Moon and Planets 68 



Part 2 Looking In: Atomic and Nuclear Physics 75 



8. 

9. 
10. 
11. 
12. 
13. 



14. 
15. 
16. 
17. 
18. 
19. 
20. 



The Atomic Idea 77 
Wave Motion 91 
Electric and Magnetic Forces 116 
The Electron 136 
Ions and Isotopes 149 
Electromagnetic Radiation 
I Classical Theory 156 
II Quantum Effects 167 
Relativity Wonderland 184 
Hydrogen Atom Bohr Model 203 
Quantum Dynamics 216 
Radioactivity 238 
Nuclear Reactions 252 
Absorption of Radiation 260 
Unconventional Energy Sources 272 



Appendices 

A. Reaction Thrust 289 

B. Burnout Velocity and Range 291 

C. Schrodinger Wave Equation 293 

D. 1. Books for a Physics Teacher's Reference Shelf 295 

2. Some Periodicals for a School Science Library 296 

3. Professional Organizations of Interest 
to the Physics Teacher 296 

4. Some Suppliers of Physics Apparatus for Teaching 297 

5. Greek Alphabet 297 

6. Symbols 298 

7. Instruction on the Use of a Slide Rule 298 
7A. Slide Rule Bibliography 300 

8. Trigonometric Functions 301 

9. Logarithms to the Base e 302 

10. Values of Physical Constants 304 

11. Conversion of Electrical Units 305 



Index 307 



PART 



Looking Out: 

Rockets, Satellites, 

Space Travel 



What's Up? 



In his millennia of looking at the stars, 
man has never found so exciting a 
challenge as the year 1957 has sud- 
denly thrust upon him. 

Fred L. Whipple 



1.1 ASTRONAUTICS TODAY 

Fictional accounts of space travel had been written before the 
lime of Jules Verne. In the second century a.i>., Lucian of Santos 
wrote of a visit to the moon. But the foundation for converting 
fantasy into an engineering possibility was the invention of lite 
rocket. Most current progress in the science and technology of 
space flight is an outgrowth of the efforts since World War II to 
develop long-range military missiles. 

Popular concern aboul space technology was aroused when 
the first artificial earth satellite was launched by the Soviet Union 
in 1!J">7. In the next 3 years, some 36 satellites carrying instru- 
ments were launched, 30 of them by the United States. Also dur- 
ing this period several moon shots and space probes sought infor- 
mation farther from the earth. Next came the spectacular manned 
orbital flights made l>y astronauts of the Soviet Union and the 
United States. 
3 



4 Looking Out: Rockets, Satellites, Space Travel 

In the "World Almanac" one may find a listing of the major 
space vehicles launched since 19">7. Table 1.1 gives the longer 
perapective of the history of man's use of rockets. 



Table 1.1 
c. 300 

c. 1200 



c. 


1780 




1792 


c. 


1800 




1812 


c 


1830 




1846 




1913 


C 


1915 




1926 


c 


1930 




1931 




1932 




1936 




1941 




1942 




1944 


c 


. 1945 




1946 




1949 


c 


. 1952 



Milestones in missilery 
Hero of Alexandria uses the reacting force of escaping steam to 
propel an experimental device. 

Chinese use gunpowder to propel "arrows of flying fire," equiva- 
lent to present-day skyrockets. 
Advanced type of rocket developed in India. 
Troops of Tipu, Sultan of Mysore, use rockets against British in 
second Mysore War. 

Sir William Congreve of Great Britain improves rocket propellant 
to provide considerable increase in range. 

British use rockets in attack on Fort McHenry (Baltimore), com- 
memorated in the line "... and the rockets' red glare" in our 
National Anthem. 

William Hale, an American, increases stability of rockets by adding 
nozzle vanes. 

Mexican War sees first use of rocket weapons by United States 
in o war. Lifesaving rockets developed by English and German 
inventors. 

Ramjet proposed and patented in France. 

World War I sees advent of guided missile to supplant aimed 
rockets. 

Dr. Robert H. Goddard, professor of physics at Clark University, 
fires first successful liquid- fuel rocket. 

Germans experiment with the pulse jet, used to power the Nazi V-1 
"bun bomb" of World War II. 
Germany uses liquid rocket fuel. 

Captain Walter Dornberger undertakes development of liquid- 
fuel rocket weapons for the German Army. 

German Peenemunde Project is organized, to develop war rockets. 
United States starts work on controllable rocket weapons. 
American Razon missile, controllable in both azimuth and range, 
is developed. 

United States government awards first contract for research and 

development of guided missile to General Electric Company. 

Germany uses V-1 buzz bomb, V-2, and other rocket missiles in 

World War II. United States uses "Weary Willie" unmanned 

bombers. 

Work is started in the United States on an intercontinental ballistic 

missile program, the MX-774, 

First flight of a missile beyond earth's atmosphere is made at 

White Sands, N. Mex. 

United States long-range missile program is stimulated by Atomic 

Energy Commission warhead developments. 



What's Up? 5 

Table 1,1 Milestones in missilery (continued) 

1954 United States starts ICBM program; USAF awards contracts to 

Convair, North American Aviation, and General Electric. 
1957 First artificial earth satellites — Sputniks I and II — launched by 
rocket (October 4 and November 3). 
1.31.58 Explorer I, first United States satellite, launched. 

3.17.58 Vanguard I, first "permanent" satellite launched by the United 
States. 

10,11.58 Pioneer I, first lunar probe, launched by the United States. 
12.18.58 Project Score (Atlas) launched, broadcasting a human voice from 

outer space for the first time. 

1.2.59 Russia launches Lunik, first satellite to orbit around sun. 

3.3.59 Pioneer IV launched, first United States satellite to orbit sun. 

9.12.59 Russia launches first space vehicle to land on moon. 

10.4.59 Russia launches first satellite to orbit moon. 

8.11.60 United Slates recovers first space vehicle from orbit. 

4.12.61 Manned orbital flight achieved in Soviet Vostok satellite, 

2.20.62 Pfoject Mercury succeeds in manned orbital flight, 
7.10.62 Telstar satellite relays first transatlantic television, programs. 
8.13.62 Two Russian astronauts put in reloted orbits. 

8.27.62 Mariner II launched to encounter Venus. 
12,14.62 Mariner II passed within 22,000 mi of Venus, reporting data on 
temperature, cloud cover, magnetic field, particles and radiation 
dosage encountered throughout voyage. 

1.2 ASTRONAUTICS TOMORROW 

What feat may be expected, perhaps in the next 2 or 3 years, 

from adaptations of the intercontinental ballistic missiles already 
available? The staff of the RAND Corporation has estimated 
that we shall be able to do the following: 

1. Orhit satellite payloads of 10,000 lb to 300 mi altitude 

2. Orbit satellite payloads of 2,500 lb at 22,000 mi 

3. Impact 3,000 lb on the moon 

4. Land, intact, more than 1,000 lb of instruments on the moon, 
Venus, or Mars 

."). Probe the atmosphere of Jupiter with 1,000 lb of instruments 
fi. Place a man, or men, in a satellite orbit around the earth for 
recovery after a few days of flight 



1.3 WHO SHOULD CARE? 

By the military, the costly development of the rocket has been 
pushed chiefly as a gunless artillery device and for bombardment 



6 Looking Out: Rockets, Satellites, Space Travel 

over intercontinental distances with nuclear warheads. Rocket- 
launched viewing satellites may make possible the inspection of 
foreign territory and thus discourage preparation for war. The 
difficulties that statesmen now have in reaching agreement on 
inspection and on disarmament will, however, probably increase 
as military space technology expands. 

In addition to the military reasons, there are many scientific 
incentives for making satellites. Some important problems await- 
ing investigation are: 

1. Determination of density, pressure, and temperature in the 
upper layers of the atmosphere 

2. Exact measurement of the dimensions of the earth, the conti- 
nental distances, and other geodetic measurements 

:i. A detailed study of radiation from the sun 

4. Observation of the intensity of cosmic rays and other radia- 
tion in the earth's atmosphere 

f). Correlation of the currents of nuclei, neutrons, and other 
particles flying toward the earth with sunspot activity 

(i. Kstimation of the distribution of mass in the earth's crust 
from the orbital planes of the artificial earth satellite 

7. Study of the propagation of radio waxes in the upper atmos- 
phere and provision of radio communication, navigation bea- 
cons, and television with the aid of satellites 

8. Improvement in weather forecasting 

it. Making feasible astro no mica I investigations without atmos- 
pheric and other disturbances 
10. Study of biological specimens in environments different from 
that on earth 

Although we have these incentives for space exploration, it 
is likely that such exploration will enlighten us in fields even 
beyond our present speculations. 

Space flight obviously demands development of devices of 
great reliability 1o operate for long periods under extreme condi- 
tions of environment. Engineering advances depend on funda- 
mental scientific knowledge, and in the past these advances have 
contributed tools for the obtaining of new knowledge. This inter- 
action or feedback is occurring in astronautics. When spaceships 
can carry instruments, or man himself, into other parts of the 
solar system, new information will surely become available for 
the physicist, biologist, and astronomer. 



What's Up? 7 

The philosopher and the theologian are already adapting their 
thinking to the eventuality that man may encounter life else- 
where in the universe. It seems probable that just as the tele- 
scope profoundly altered seventeenth-century thought, the space 
vehicle will extend twentieth-century man's understanding of the 
universe and his role in it. 

1.4 CAREERS IN ASTRONAUTICS 

Astronautics touches almost all fields of current science and tech- 
nology. 1 1 may be expected to lead to entirely new fields. Entrance 
into the field of astronautics can be made by one who has acquired 
knowledge in one or more of these fields: mechanical, aeronaut- 
ical, and electrical engineering; mathematics; physics; biophys- 
ics; and chemistry. Mathematics and physics are basic. With 
nuclear power a necessity for distant space travel, the field of 
nuclear physics is of special importance. 

Information about careers in astronautics can be obtained 
from the corporations active in this field, and also from the agency 
which coordinates the government's activities, NASA. The .Na- 
tional Aeronautics and Space Administration was created by an 
act of Congress signed by President Eisenhower on July 2H, I9">8. 
The act declared that "it is the policy of the United States that 
activities in space should be devoted to peaceful purposes for the 
benefit of mankind." 

FILMS ON ASTRONAUTICS 

For a listing of some 9(1 films on rockets, missiles, and space science see 
It. L Weber: Films for Students of Physics, Supplement I, American 
Jcwntd of I'h units, SO: 321 327 (19«2). 



SUGGESTIONS FOR FURTHER READING IN ASTRONAUTICS 

Hooks: 

Adams, Carsbie C: "Space Flight, " McGraw-Hill Book Company, Inc., 
New York, I95S, 373 pp. 

Hcnson, t). 0., and H. Strugliold: "Physics and Medicine of the Atmos- 
phere ami Space," .John Wiley A Sons, Inc., New Yolk. IWiO, 
•»4f) pp. 

"uchheim. It. W„ and Staff of HAND Corp.: "Space Handbook," 
House Document 80, U.S. Government Priming Office, Washing- 



8 Looking Out: Rockets, Satellites, Space Travel 

ton 25, D.C, 1959. Also Random House, hit-. (Modern Library 

edition), New York. 
The National Aeronautic* and Space Administration: "Space: The 

New Frontier," U.S. Government Printing Office, Washington 25, 

D.C., 1962,48 pp. 
Ordway, Frederick I.: "Annotated Bibliography of Spaee Science and 

Technology," Arfbf Publications, P.O. Box 6285, Washing! on 15, 

D.C., 1011-2. 
Scifert, Howard S. (ed.): "Sparc Technology," .John Wiley & Sons, Int.. 

New York, 1959. Text based on graduate-level lectures presented 

by University Extension, University of California, in cooperation 

with Hamo-Wooldridgc Corp. 

Periodicals: 

Astronautics. Published monthly by The American Pocket Society, 

Inc., 500 Fifth Ave, New York W. NY. 
Aviation HVefc (Including Space Technology). McGraw-Hill Publishing 

Company, Inc., 330 West 42 St., New York 36, N.Y. 
Missiles and Rockets. Published weekly by American Aviation Publi- 
cations, 1001 Vermont Ave., NW, Washington 5, D.C. 
Sky and Telescope. Published monthly by Sky Publishing Co., 60 Garden 

St., Cambridge 38, Mass. 
Space Aeronautics. Published monthly by Conover-Mast Publications. 

foe., 20.") East 42 St., New York 1 7, N.Y. 
Space Age. Published quarterly bv Quinn Publishing Co., Kingston, 

N.Y. 
Spaceflight. Published bimonthly hy British Interplanetary Society, 

12 Bessborough Gardens, London, SW 1. England. 
Space Journal. Published quarterly by Space Enterprises, Inc., P. O. 

Box 94, Nashville, Tenn. 

Publications of Sperial Interest to Students: 

Adams. Carsbie C, Wernher von Braun, and Frederick I. Ordway: 
"Careers in Astronautics ami Rocketry," McGraw-Hill Book Com- 
pany, Inc., New York. 1902, 248 pp. 

Map of the Moon, chart, 45 by 35 in.. General Electric Missile and 
Space Vehicle Department, Valley Forge Space Technology Cen- 
ter (Mail: P.O. Box 8555, Philadelphia 1. Pa.). 

Map of Outer Spaee, chart, 28 by 25 in., General Electric Missile and 
Space Vehicle Department. 

"Short Glossary of Space Terms," National Aeronautics and Spaee 
Administration, U.S. Government Printing Office, Washington 25, 
D.C, 1962, 57 pp. 

"Space Primer: An Introduction to Astronautics," Convair-Asfro- 
nautics. Dcpl. 120, P.O. Box 112s, San Diego 12, Calif., 72 pp. 



Environment 
of Space 



The most incomprehensible thing 
about the universe is that it is compre- 
hensible. Albert Einstein 



2.1 INTRODUCTION 

Our sun, the 9 major planets, 31 known moons, and thousands 
of lesser bodies all revolving around the sun constitute the solar 
-v-ii'iii. The planets move around the sun in the same direction 
in elliptical orbits which are nearly circular (big. 2.1). All the 
orbits lie in nearly the ecliptic plane of the sun's apparent path 
among the stars. The orbit of Pluto deviates most, about 17°. 
from the ecliptic plane. The zone about 17° wide on each side of 
the ecliptic plane is known as the zodiac. 

The average distance of the earth from the sun is 92,900,000 
mi, a distance which is defined as one astronomical unit (a.u.). 
The diameter of the orbit of Pluto, the outermost member of the 
solar system, is about 79 a.u. 

The four inner planets, Mercury, Venus, Earth, and Mars, 
are sometimes called the terrestrial planets, They are relatively 
9 



10 Looking Out: Rockets, Satellites, Space Travel 

small, dense bodies. The next four outer planets, Jupiter, Saturn, 
Uranus, and Neptune, are called the major planets or the giant 
planets. They are relatively large bodies with ice and rock cores 



% >. Meteors 



Jupiter 







Fig. 2.1 Solar system; orbits of Mercury, Venus, Eorth, Mars, and 
Jupiter. 

below their visible atmospheres. Physical data on objects of prin- 
cipal interest in the solar system are given in Table 2.1. 

Table 2.1 Physical data on some bodies in the solar system 



Body 


Mean 


Man, 


Diameter, 


Gravita- 


Intensity 


Length 


Length 




distance 


times 


mi 


tional force 


of 


of day 


of year 




from 


earth's 




at solid 


sunlight, 








tun, 


mass 




surface. 


rel. to 








a.u. 






9'« 


earth 






Sun 




329,000 


864,000 


- 








Mercury 


0.39 


0.05 


3,100 


0.3 


6.7 


68 d 


88 d 


Venus 


0.72 


0.82 


7,500 


0.91 


1.9 


? 


225 d 


Earth 


1 


1 


7,920 


1 


1 


24 hr 


365 d 


Mars 


1.52 


0.11 


4,150 


0.38 


0.43 


24.6 hr 


1.9 yr 


Jupiter 


5.2 


317 


87,000 


t 


0.037 


10 hr 


12 yr 


Saturn 


9.5 


95 


71,500 


t 


0.01 1 


10 hr 


29 yr 


Uranus 


19.2 


15 


32,000 


t 


0.0027 


11 hr 


84 yr 


Neptune 


30 


17 


31,000 


t 


0.0011 


16 hr 


165 yr 


Pluto 


79 


0.8 


? 


| 


0.0006 


? 


248 yr 


Moon 


1.0 


0.012 


2,160 


0.17 


1 


27 d 





* Hoi no solid surface. 

f location of solid surfoce not known (far below dense atmospheric gases). 



2.2 THE SUN 

The sun, whose gravitational attraction chiefly controls the mo- 
tion of planets in the solar system, is classified as about average 



Environment of Space 11 

among stars in size, in temperature, and in brightness (spectral 
type C-2). Its nearness to the earth makes the sun appear to us 
very large and bright. The surface temperature of the sun has 
Keen measured as about ti000°C, or I0,000°r', The energy output 
of the sun as light and heat is remarkably constant. Solar energy 
arrives at the surface of the earth at an average rate of 1.35 kilo- 
watts/ m-. This solar energy, resulting from a series of thermo- 
nuclear reactions, makes life possible on the earth. 

Sunlight takes a little more than 8 min to reach the earth. 
When analyzed with a spectrograph, sunlight is found to consist 
of a continuous spectrum, but with the colors crossed by many 
dark lines. The absorption lines are produced by gaseous materials 
in the atmosphere of the sun. From their lines, some 70 of the 
chemical elements occurring on earth have been identified as 
present in the sun. 

The radiating surface of the sun is called the photosphere. 
Above it is the chromosphere, visible to the unaided eye al times 
of total eclipses as a turbulent pinkish-violet layer. The pearly 
light of corona extends millions of miles beyond the chromo- 
sphere. Corona are related to the appearance of sunspots — dark, 
irregular regions which may extend several hundred thousand 
miles across and whicii may last for a few weeks to several months. 

The output of ultraviolet radiation, radio waves ("static"), 
and charged particles (cosmic rays) from the sun is highly 
variable. 



2.3 THE PLANETS 

because of its nearness to the sun, Mercury is difficult to observe 
ami knowledge of its physical characteristics is not very accurate. 
Mercury has a mass about one-twentieth the mass of the earth. It 
lias no moon. .Mercury has a rockysurl'ace. probably similarlof hat 
»f our moon. Mercury always keeps the same side turned toward 
the sun. This side probably has surface temperatures as high as 
b)()°C, while the side in darkness is cold enough to retain frozen 
gases, with temperatures approaching absolute zero. 

Venus is slightly smaller than the earth, shrouded in a dense 
; ilinospherc opaque to light of all wavelengths, Neither free oxy- 
Ren nor water vapor has been detected on Venus. Carbon dioxide 
] s abundant in its atmosphere, with nitrogen and argon also 



12 Looking Out: Rockets, Satellites, Space Travel 

present. It is thought that die surface of Venus is hot {about 
425°C), dry, and dark beneath a continuous dust storm. 

Mars lias u diameter ahoul half that of the earth, its rate of 
revolution is about the same as that of earth, and its axis, too, is 
inclined about 2;">° from the plane of its orbit. Mars takes <»87 of 
our days to make one circuit of the sun. Although the orbit is 
nearly circular, it is not centered on the sun; Mars is more than 
30 million mi farther from the sun at some parts of its year than 
at others. More than half of the surface of Mars is a desert of 
rusty rock, sand, and soil. The rest of the sin-face shows seasonal 
color changes which have been interpreted as due to vegetation. 
While noon summer temperature on Mars probably reaches 30°C, 
night temperatures probably fall to — 70°C. The atmosphere 
{mostly nitrogen) on Mars has a pressure about 10 per cent of 
the earth's atmosphere. Oxygen has not been detected. The white 
polar caps are probably frost layers, which on melting furnish 
moisture for the summer growth of vegetation. Mars has two 
small satellites about 5 and 10 mi in diameter. 

Each of the four giant planets, Jupiter, Saturn, Uranus, and 
Neptune, seems to have a dense rocky core surrounded by a thick 
layer of ice and covered by thousands of miles of compressed 
hydrogen and helium with smaller quantities of methane and 
ammonia. These planets receive relatively feeble radiation from 
the sun, so that the temperatures of their upper atmospheres 
range from -100 to -200°C. These planets rotate rapidly and 
in the same direction. Some of the satellites of these planets are 
larger than the earth's moon and may have physical character- 
istics less formidable for space-flight visits than the major planets 
themselves. Jupiter has 12 satellites. Four are bright enough to 
be easily visible with binoculars, and their rapid motion causes 
interesting changes in position from night to night. 

Saturn is the farthest of the planets visible to the unaided eye. 
It has nine satellites. Saturn is surrounded by remarkable flat 
rings in the plane of the equator. It has been suggested that the 
rings are made up of tiny particles of a shattered tenth satellite. 

2.4 THE MOON 

The moon has a mass about B V that of the earth, a diameter of 
about 2,l(iO mi, and an elliptic orbit which gives it an average 



Environment of Space 13 

distance from the earth of 239,000 mi. The moon has no appre- 
ciable atmosphere. Its surface, comprising many craters and high 
mountains, is probably dry, dust-covered rack. The moon rotates 
on its axis in a period of time equal to the period of its revolution 
about the earth, 27.3 days. The moon's elliptic orbit and its 
variation in altitude from season to season permit us to examine 
about 00 per cent of its surface, over a period of time. 

2.5 ASTEROIDS, COMETS, AND METEORITES 

Asteroids are pieces of planetlike material which, unlike the 
planets, are of irregular shape. They may be the shattered frag- 



Fig. 2.2 Radiation pressure which 
forces the tail of a comet away from 
the sun might propel a spaceship. 









* — V 

meats of one or more planets. Most measure a few miles across; 
the largest, Ceres, is nearly f>00 mi across. The orbits of most 
asieroids lie between the orbits of Mars am! Jupiter {Fig. 2.1). 
The time for one revolution varies greatly among the asteroids. 

Comets are large, loose collections of material that penetrate 
the inner regions of the solar system from outer space. The most 
famous one, named after Halley, has been sighted every 7(i.02 
years since 240 h.c, but not all return periodically. Comets have 
a head and tail. The head is made up of heavy particles and is 
attracted by the sun. The tail is made up of dust and gas and is 
forced away from the sun by radiation pressure as the comet 
sweeps past the sun. The brightness of the comets is probably due 
to reflected sunlight. The earth has passed through the tails of 
many comets without effect. 

Some 2,000 tons of material from outer space reaches the 
Ruth's atmosphere each day in the form of meteorite particles, 
1'hose enter the earth's atmosphere with speeds of 10 to 50 mi/sec 
and are heated to incandescence, producing the light streaks 
called incteors. Reflection of radio waves from the ionized paths 



14 Looking Out: Rockets, Satellites, Space Travel 

and observations of sky gk)W at twilight as well as direct visual 
counting of meteors indicate tliat a large amount of material 
is received daily, but data are inadequate. How much meteoi- 
itie material a space vehicle might encounter is an important 

unknown. 

In the night sky a faint tapered band of zodiacal light can 
|„, ..en and traced photoelectrically. It is evidence of cosmic 
dust, mierometeorites, concentrated toward the plane of the solar 
system. 

2.6 RADIATION 

Beyond the shelter of the earth's atmosphere, x rays, ultravinl.t 
rays, and cosmic rays exist at intensities which may have to be 
considered in planning exploration by space vehicles. The WA 
Explorer satellite detected an encircling belt of high-energy radMr 
tion extending upward from a height of a few hundred miles, most 
intense in the equatorial region (Fig. 2.3). The earth's magnetic 
field traps the particles, chiefly electrons and protons, constituting 
the radiation belt. During solar flares, the sun delivers as much 
as 1,000 times its normal radiation. 

The nature of the radiation found in space is described in 
Table 2.2. In order to avoid subjecting astronaut* to radiation 



Table 2.2 Radiation in space 



Name 



Nature of Charge Mass 

radiation 



Photon 


Electronic 


gnetic 








Quantum 


Electronic 


gnetic 








Xray 


Electromagnetic 








Gamma ray 


Electromagnetic 








Electron 


Particle 




— a 


lm, 


Proton 


Particle 




+■ 


1,840m, 
or 1 omu 


Neutron 


Porticle 







1,841m, 


Alpha porticle 


Particle 




+ 2. 


4 amu 


Heovy primaries 


Porticle 




&+3e 


' 6 amu 



Where found 



Radiation belts, solar radia- 
tion (produced by nuclear 
reactions and by stopping 
electrons) 

Radiation belt 

Cosmic rays, inner rodiotion 
belts, solar cosmic rays 

Vicinity of planets and sun 
(produced in nuclear inter* 
actions — decoys into pro- 
ton and electron) 

Cosmic roys (nucleus of 

helium atom) 
Cosmic rays (nuclei of heav- 
ier otomsl 



From H. E. Newell and J. E Naugle, Science, 132i 1*65 (1960). 




Fig. 2.3 Space radiation: cross section of the radiation pattern in longitude 
75° west, from Explorer satellites. (Adapted from the New York Times, 
October 27, 1958.) 




'9- 2,4 Pioneer V, o 95-1 b highly instrumented space planetoid, was launched 
March 11, 1960, to supply the first comprehensive data collected in interplane- 
,ar y space. 

15 



o 

t— 

< 



3 

o 



z 
o 

I— 

u 

u% 

in 

1/1 

§ 



z 

I 

s 



z 




■■■'. : '" v -. 



'■ ' /.-■■■ ■ ■ ■ >. 

'■-,. 
i ■■■ ■ ■■■■*■■-*•/■■ ■ 



•a c 

y, 5 

S-5! 






£ 5 



-n. Dl 

£ 1 

* t 

I § 

1) 

* a 






«i - 

. J! 

i & 




Environment of Space 17 

in excess of tolerable dosages (Table 2.3), it may be necessary 
to plan flights from the earth along trajectories which avoid the 
regions of concentrated radiation (Fig, 2,">). 

Table 2.3 Maximum permissible radiation dosages and some typical 
exposure levels {in roentgens) 



Item 



Amount 



Permissib/e exposures 
Maximum permissible dosages 

Maximum permissible emergency exposure 

Typical exposures 
Normal radiation level (sea level) 
Undisturbed interplanetary space (cosmic rays) 
Heart of inner belt (protons) 
Heart of outer belt (soft x rays) 
So!ar proton event (protons) 
Total exposure 



0.3* r/'quarter 
5.0 r/yr 
25 r 

0.001 r day 
5-12 r/yr 
24 r/hr 
~200 r/hr 
10-10" r/hr 
2-400 r 



E Ei 



* Limit prescribed for radiation workers. Under this limit the yearly 
maximum would be 1.2 r. 

From H. E. Newell and J. E. Naugle, Science, 132: 1465 (1960). 

2.7 MORE DISTANT SPACE 

The sun's nearest star neighbor is Alpha Centauri, which is more 
than 4 light-years away. (In contrast, the outermost planet of 
the solar system, Pluto, is only 0.000 light-year from the sun.) 
The relative brightness of several stars and their distances from 
the sun are shown in Table 2.4. 

Insofar as man knows, the universe is infinite. Scattered 
throughout this void is an apparently endless number of galaxies, 
each of which contains millions of stars. Some galaxies are them- 
selves grouped in clusters. The constellation Corona Borealis is 
made up of some 400 galaxies. In the observable region around 
us there are an estimated I billion galaxies, with an average dis- 
tance between galaxies of about 2 million liglil -years. 

Galaxies usually have the shape of disks thousands of light- 
years in diameter. The larger galaxies have spiral arms suggesting 
a pattern of rotation. Our own galaxy, the Milky Way, appears 
tn have this form. Our solar system is believed to be situated in 



16 



18 Looking Out: Rockets, Satellites, Space Travel 
Table 2.4 Some star distances 



Star 



Brightness, 
relative to sun 



Distance from sun, 
light-yeors 



Alpha Centauri 
Barnard's Star 
Wolf 539 
Sirius A,B 
Proeyon A 
Altair 
Argo 
Deneb 
Betelgeuse 



1 

1 60,000 

23 

6 

8 

5,200 

6,600 

13,000 



4.3 
6.0 
7.7 
8.7 

11.3 
16.5 

180 
640 
650 



one of the spiral arm*, about 30,000 light-years from the center. 
The solar system is moving at a speed of about 1-.0 mi see, but 
it takes 200 million vears to complete one circuit of the, galaxy. 

Ylthough presently envisioned techniques may lead to 
manned exploration in the solar system, they will not suffice for 
exploring (he vast distances beyond. 

If as has been estimated, less than 12 per cent of all stars 
have planetary systems, then nut of some 200 billion stars m our 
galaxy, there are some billion with planetary systems. One is led 
to speculate that out of this number there are probably some 
systems with earthlike planets that may support life. Communi- 
cation with distant planets of the galaxy is a matter «,t specula- 
tion only. And bevond our galaxy arc other galaxies at least out 
to the limits accessible to present telescopes: some '2 billion light- 
years in all directions. 

A FILM INTRODUCTION TO SPACE 

Ut*mB.28wto (I960), National Fiim^r,Ur Ca^la. For vent from 

Contemporary Films. Inc., 287 West 25 St, New York I, N.Y. 

Teachers' guide available, 

SUGGESTIONS FOR FURTHER READING 

Baker, Robert H.: "Astronomy," 7th e<L, I). Van Nostrum! Company, 

Inc., Princeton, NJ. t 1989. „ „., 

Bauer Carl \ : "The Universe bom fee Known to the Unknown, n» 

Pennsylvania Stale University. Continuing Kdueation, Umvcrsitj 



Environment of Space 19 

Park, Pa., 1982, 54 pp. A manual for adult discussion study groups 

to which a Guide for the Discussion Leader is keyed, 
Duncan, John C: ■■Astronomy," 5th ed., Harper it' Row, Publishers, 

Inc.. Now York, 1055. 
Hoyie, Fred: "Frontiers of Astronomy." New American Library of 

Win-id Literature, Inc., New York. 1955. 
James. .1. N.: The Voyage of Mariner II, Scientific American, 209: 

70-84 (1983). 
Krogdfthl, Wasley S.: "The Astronoinical [.'inverse," The Macmillan 

Company, New York, 1952. 
McLaughlin, Dean II.: "Introduction to Astronomy," Houghton 

Mifflin Company, Boston, 1981. 
Newell, II. !■",.. and J. K, Xaugle: Uadiulion Environment in Space. 

Sriri„r. 1:12:1 if 15 I 172 (Mil ill). 

Shaw, John II.: The Radiation Environment of Interplanetarv Space, 

Applied Optics, I -. 87-95 (1902). 
Zim. Herbert, and Robert 1L linker: "Stars." Colder, Press. Xew York 

1950. 

QUESTIONS FOR DISCUSSION 

1. Would you expect Mercury to have an atmosphere, that is, a 
permanent gaseous envelope? 

2. Estimate the fraction of the total mass of the solar system which 
» i" I lie sun. Am. (iO.fi per cent 

3. Does a physical environment of the sort needed to support plant 
and animal life such as we know exist elsewhere in the universe? Where? 
Can you conceive of a form of life not based on water chemistry? 
Might ammonia or fluorine compounds serve? Where in the universe 
would you expect lids differenl form of life to exist? If it docs exist in 
intelligent. r,, rmi ,[,, vnu tM j n k Wl , ( . ou j ( | ,. tHnmu „i t . at p ,vilh it? 

I. A point on the earth's equnlor is carried ahoul 1,090 uii.hr by 
the rotation or the earth. Jupiter has an equatorial diameter II times 
thai of the earth and a day of 10 hr. Calculate (he speed of a point on 
the equator of Jupiter. .W _>2.i mi hr 

5. Express the diameter of Pluto's orbil in mi. 

Ami. 7,309 million mi 

6. In what ways is it true that all our sources of energy— plant life, 
"oal. oil, and water -arc derived from the radiant energy we receive 
uom the sun? 

T. Furnish some evidence for or against the statements: The climate 
"ii Mars is similar to that which one would encounter on a iO-mi-high 
' ' -'i on earth. Granted transportation, a self-sustaining colon v might 
'"' established on Mars. 

8. The four outermost of Jupiter's 12 satellites revolve about 

• npitei- rrom east to west, contrary to Ihe motion of most satellites in 

i solar system and to the direction of revolution of Ihe planets around 

1 «m. Can yon suggest a possible reason for this retrograde motion? 



Rocket Propulsion 21 



Rocket 
Propulsion 



Necessity is not the mother of inven- 
tion; knowledge and experiment are 
its parents. W. R. Whitney 



All vehicles move !>y reaction with some other thing. Cars require 
traction on the road. Snip* and planes push or pull themselves 
through water or air. Only rockets carry their own "something 
else" to push against. In the words of J. X. Savage, "a rocket is 
any machine that propels itself by ejecting material brought 
along for the purpose." 

A rocket is an internal-combustion engine that carries its own 
supply of oxygen (in any of several forms of "oxidizer")- There- 
fore, it does not require air but can operate in a vacuum, as in 
space. 

The description of a rocket in flight is a particular application 
of the general theory of the dynamics of rigid bodies. It is con- 
venient to consider separately the motion of the center of mass 
and the motion of the body around its center of mass. The former 
is the theory of flight performance, the latter, the theory of sta- 
bility and control. The powered flight of a ballistic rocket is usu- 

20 



ally, for practical reasons, confined to two dimensions. So the 
theory of motion in one plane is adequate. In further simplifi- 
cation, we may begin by considering the flight path to be a 
straight line (one-dimensional theory). 

We shall consider in this chapter the basic principles of rocket 
propulsion; the effects of mass ratio, specific impulse, and 1 1 mist/ 
weight ratio on the flight of a single-stage rocket; and then the 
performance of multistage rockets. 



3,1 MECHANICAL PRINCIPLES: NEWTON'S LAWS 

Three laws formulated by Sir Isaac Newton in the seventeenth 
century are fundamental to rocketry: 

1. A body at rest remains at rest and a body in motion continues 
to move at constant speed in a straight line unless acted upon 
by an externa}, unbalanced force. 

2. An unbalanced force acting on a body produces an acceleration 
in the direction of the net force, an acceleration that is directly 
proportional to the force and inversely proportional to the 
mass of the body. 

3. For every force that acts on one body, there is a force equal 
in magnitude but opposite in direction that reacts upon a 
second hody. 

Table 3.1 Consistent systems of units for Newton's second law 



Name of system 


Unit of mass 


Unit of force 


Unit of 










acceleration 


Mks (absolute) 


kilogram 


newton* 




meter, second - 


Cgs absolute 


gram 


dyne* 




centimeter second - 


Cgs gravitational 


No name assigned 
m = W, g 


gram 




centimeter, second 3 


British absolute 


pound 


poundal* 




foot/second 2 


British gravitational 


slug* 


pound 




foot /second 1 


Any system 


W, g 


Same unit as that 


Same unit as that 






used for 


W 


used for g 



In each set the starred unit is the one usually defined from the second law 
so as to make k = 1 in F = kma. 



22 Looking Out : Rockets, Satellites, Space Travel 

3.2 UNIFORMLY ACCELERATED MOTION 

It is convenient to list and remember the equations which apply 
to a body which moves with constant acceleration in a straight 
line. This is a special case, but one often met. The average speed 
v is the distance traveled divided by the time required, 5 = s/t, or 

, = U M> 

Since we have assumed motion in a constant direction, the accel- 
eration is the change in speed divided by the time, a - (»i — Vi)/t, 
or 

v, - „ = at 03) 

Since the speed changes at n uniform rate, the average speed f is 
equal to half the sum of the initial and final speeds: 

»! + "« (3.3) 

2 

By combining these, two other useful equations can be obtained. 
Eliminating u s and v, we get 

s - M + Jo* 1 < 3 - 4 > 

By eliminating 6 and t from Eq. (3,1) to (8.3), we get 



S = 



3.3 REACTION PRINCIPLE 

A rocket engine develops thrust by employing Xewton's third 
law in the following manner. Imagine a stationary sphere (Fig. 
3.1«) containing a combustible mixture of gasoline vapor and air. 

p F 

(a) (6) M 

Fig. 3.1 Reaction, the principle of jet propulsion. 




a Ah- P, 



Rocket Propulsion 23 

If this mixture were ignited, there would result a high pressure 
p t in the chamber exerting force equally in all directions. The 
sphere would remain at rest because there would be no net force 
acting on it. 

Consider a section to be removed from one side of the sphere 
so that the gases could escape. The sphere would now experience 
a net force. Since there would be no balancing force across area 
A\ (Fig. 3.1b), the force on area A-> would cause the sphere to 
move to the left. The magnitude of this force or thrust F would 
be equal to the product of the pressure p c in the chamber and the 
area A t of the throat: 



F = pAt 



cm 



A greater force can be obtained under certain conditions (Sec. 
3.(>) by adding an expansion nozzle at the exit (Fig. 3.1c). The 
contribution of the nozzle is represented by a thrust coefficient 
Cp used as a multiplier in the previous thrust equation, so 



F = PrAtCp 



(3.7) 



From Xewton's laws, if F is the net external force applied to 
a system, the rate of change of momentum of the system is 



A(mi>) _ „ 



(3.8) 



When a rocket is in free space, the net external force acting upon 
the rocket is zero. If mass particles are ejected from the rocket 
with a constant exhaust velocity iv, their rate of change of momen- 
tum gives the rocket an accelerating force 



' - " A, * 



(3.9) 



The negative sign expresses the fact that F and c, are in opposite 
directions. 

The exit pressure p„ of the gas from a rocket often may be 
either greater or lower than pressure of the racket's environment, 
ambient pressure p«. Also, while p r remains constant, p„ will de- 
crease as the missile gains altitude. If the difference between the 
two pressures is multiplied by the exit area .4,.. we have the mag- 
nitude of the unbalanced force (p c — p a )A e acting on the rocket. 



24 Looking Out: Rockets, Satellites, Space Travel 

This force is called the "pressure thrust," in contrast with the 
"momentum thrust" expressed in Eq. (8.9). 

The total thrust of a rocket engine can be expressed as the 
sum of the momentum thrust and the pressure thrust: 



F = - -^T- V* + (}Je - V-) A - 



(3.10) 



"Usually, the only term on the right-hand side of this equation 
that will vary with respect to time is p„, the ambient pressure. 



66 
64 


/-" 


. 62 


S 


o 


f 






X 


X 


=e 60 


/ 






1 


/ 


£ 58 


- / 


56 

54 


i il _! 



20,000 40,000 60,000 

Altitude, ft 



80,000 



100,000 



Fig. 3.2 Decrease of thrust with altitude, V-2 missile. 

Positive thrust Negative thrust 




-t 



Fig. 3.3 Pressure thrust in a rocket motor. 

The resulting decrease in thrust as a missile gains altitude is 
shown in Fig. 3.2. If, as often at sea level, p e < P-, the pressure 
thrust term will be negative (Fig. 3.3). 



Rocket Propulsion 25 



3.4 EFFECTIVE EXHAUST VELOCITY 



In order to simplify the thrust equation, an effective exhaust 
velocity >\. fi is defined as 

F 

V,n Am A! 

Then Eq. (3.10) is written in simplified form as 



(3.11) 



_ Am 



(3.11a) 



Of course v M is variable with altitude, whereas », is constant for 
a particular rocket system. Cnder optimum conditions for expan- 
sion of the gas, when p, = p„, the effective exhaust velocity e,. f r 
if equal to the theoretical exhaust velocity r,. 



3.5 SPECIFIC IMPULSE 



The performance of a rocket engine is conveniently described by 
its specific impulse. This is the thrust produced divided by the 
weight of propellant consumed per second 



Aw/At 



(3.12) 



Since F and w are expressible in the same unit (e.g., pounds), the 
unit for specific impulse is the second. If other factors are held 
constant, the speed that a missile can attain is directly propor- 
tional to the specific impulse of its propellants. 

The specific impulse varies with altitude, since thrust is vari- 
able with altitude. By combining Kqs. (3.11) and (3.12), the 
following useful relationship is obtained: 



j _ /■' _ V v it Am / At r,. fi 
Air/ At " i) Am II ' ~f 



(3.13) 



To avoid the difficulty of having /, become infinitely large as the 
gravitational acceleration g approaches zero at high altitudes, it 
is generally agreed that the value g u = 32.2 ft/sec- si mil be used 
in K<). (3.13): 



i t — — 



(3-14) 



26 Looking Out: Rockets, Satellites, Space Travel 



The simplified graph (Kig. 8,4) b intended to summarize the 
facts that certain quantities, such as theoretical exhaust velocity 
u„ propcllant flow rate Am /At, gas pressure in chamber p„ and 
exit pressure p, are constant for the rocket system. Other quanti- 
ties, such as thrust F, effective exhaust velocity v Mt and specific 



F> "•«' 4 




-*£.v e ,P e ,P c 



Sea level 



Altitude 



150,000 ft 



Fig. 3.4 Simplified representation of the fact that dm, dt, v„ p„ and 
p t are independent of a rocket altitude, while F, v,ti, and (, increase 
with altitude since they depend on ambient pressure p a . 

impulse /„, vary with altitude since they depend upon the ambient 
pressure p a . 

Table 3.2 Performance of typical liquid propellant combinations (calculated 
for expansion from 300 lb in. 1 to 1 otm) 



Propellant combination 



Mixture Exhaust Specific 

ratio velocity, impulse, 

(oxidizer/fuel) ft/sec sec 



Liquid oxygen and 75% ethyl 


alcohol, 








25% H 5 (V-2 propellant) 




1.3 


7700 


239 


Liquid oxygen and liquid H-,. 




5.33 


10,800 


335 


Liquid oxygen and kerosene 




2,2 


7,970 


248 


Fluorine and hydrazine 




1.9 


9,610 


299 


H-.0;:(S7%) and H,.0 (13%) 






4,060 


126 


Red fuming nitric acid and aniline 


3.0 


7,090 


221 


Ni from ethane 






7,010 


218 



From G, P. Sutton, "Rocket Propulsion Elements," John Wiley &. Sons, Inc., 
New York, 1949. 



Rocket Propulsion 27 

As the measure of over-all engine performance, specific impulse 
is related to both combustion performance and expansion per- 
formance. From thermodynamics il may be shown that 



<-£ 



(3.15) 



where T is the combustion temperature and M is the molecular 
mass of the exhaust gas. Thus a hot, lightweight gas gives a high 
specific impulse. Roth a large value for the heat of propellant 
combustion and low specific heat of the gas are desirable to pro- 
duce the high temperature. The requirement of low-molecular 
mass suggests that the products of combustion should be rich 
in hydrogen compounds. 

3.6 FUEL COMBUSTION AND EXPANSION 

The basic principles we have been discussing and some refine- 
ments in design can be illustrated by considering a typical rocket 
engine using a liquid fuel and oxidizer (Kig. 3.5). The engine con- 
Combustion De Laval 
chamber nozzle 




Fig, 3.5 Simplified liquid racket motor. 



verts the thermochemical potential energy of the propellants into 
the kinetic energy of the gas in the exhaust jet. The steps involved 
;nc propellant feed, injection, ignition, combustion, and expan- 
sion. Tin: liquid propellants arc forced from their tanks into the 
injector by means of compressed gas or a turbopump. The injector 
distributes the fuel and oxidizer in a flow pattern that causes 
thorough mixing. Ignition is started by a device at the surface 
of the injector; thereafter heat from the combustion gases main- 



28 Looking Out: Rockets, Satellites, Space Travel 

tains continuous ignition. Combustion takes place throughout the 
combustion chamber with some residual burning in the exhaust 

gas jet. 

During combustion, the propellants change from liquid to 
gas, and by electron sharing they combine to make new chemical 
compounds. Chemical potential energy is converted into thermal 
energy, raising the gas to a very high temperature. The change 
from the liquid to the gas state plus the high temperature of the 
gas results in a high chamber pressure. Gas particles are forced 
to the rear. 

It is the purpose of the nozzle to allow the gases to leave the 
rocket in smooth flow and also to accelerate these gases. The rear 
of the combustion chamber first converges to a throat area A t 
and then expands to an exit area A m which may have about the 
same diameter as the combustion chamber. The change from 
potential energy (nondireeted thermal motion of gas atoms) to 
the kinetic energy of a high-velocity gas jet occurs in two steps. 
As the gas passes through the converging portion of the nozzle, 
the decreasing cross-sectional area causes the flow to speed up. 
The gas flow reaches a maximum speed corresponding to sonic 
speed (Mach 1)* at the nozzle throat provided the chamber 
pressure exceeds a critical value, approximately twice the sur- 
rounding atmospheric pressure. The addition of a diverging nozzle 
provides for even more acceleration of the gases. A typical throat 
speed may be 4,000 ft/sec and exit speed 7,000 ft/sec. The expan- 
sion area rat in 



A, 
A t 



Cilo) 



is chosen for a particular engine to give the highest average 
thrust over the powered portion of the trajectory, For a given c 
a bell nozzle may be some 30 per cent shorter than a conical 
nozzle, and hence its use conserves rocket weight. 

An interesting phenomenon called jet separation may add 
additional thrust. When the exit pressure is very low in compari- 
son with the ambient pressure, gas flow breaks away from the 

* Mach number M is defined as the ratio of Free stream speed v to 
the local speed of sound a, M = v/a\ it is the ratio of directed molecular 
motion to random molecular motion. 



Rocket Propulsion 29 

wall before reaching the nozzle exit. The thrust coedieient is 
slightly higher during separation than for a full-flowing nozzle. 



Jet separation . 




Optimum 

expansion 




Jet separation 



Under- 

expansion 
Pe > Pa ' 



expansion 
Pe <Pa 



Expansion area ratio, 6 

Fig. 3.6 Jet separation. 

3.7 BURNOUT VELOCITY AND RANGE 

Consider the case of a rocket moving in a straight line inclined at 
an angle 8 with respect to the direction of gravity, with thrust F 
paraDel to the path. The equation of motion will be 

to 



m 



At 



— F — D + mg cos 



(3.17) 



where D is the aerodynamic drag and g is the acceleration of 
gravity at the location of the rocket. Since D usually depends on 




Fig. 3,7 Forces on a rocket, 

the shape and speed of the rocket and the density of the snrround- 
n»g air, let us assume for this illustration that the rocket is at such 



30 Looking Out: Rockets, Satellites, Space Travel 

high altitude that D = 0. If we divide Eq. (3.17) by m and use 
Eq. (3.13) to set F = gJ.Am/At, we have 

1 Am 



t'^'irs- 



(3.18) 



If we assume tliat the rocket starts from rest, v = 0. We set 
/« = tVi/ff and let R he the ratio of initial mass to final burnout 
mass, R = m u /m b . Then E(|. (3.18) can be solved (Appendix ») 
to find the velocity a at burnout 

vt = hu In R — gh cos (3.19) 

where k is the duration of burning. The two averages iv« and g 
are necessary since the values of both effective exhaust velocity 
and gravitational acceleration are dependent on altitude. 

Greater range and less time for interception of a rocket will 
result from increasing the burnout velocity of the missile. This 
improvement, can he obtained, according to Eq. (3.19), by 
increasing the effective exhaust velocity and the mass ratio. 



3.8 MASS RATIO 

The mass ratio is defined as the quotient of the initial or total 
mass m of a rocket and its burnout mass »h 4 : 



III:, 



(3.20) 



This is one or several dimension less ratios useful in comparing 
rocket designs. Others, whose- definitions should lie apparent, arc 
the thrust, full-weight ratio, the dead-weight fraction, and the 
payload fraction. The burnout mass is related to the initial mass 
simply by 

(3.21) 



Am , 



where Am /At is the propellant flow rate, 

From Eqs. (3.18) and (3.19) it is evident that to achieve the 
desirable high burnout velocity, a fuel with high specific impulse 
is needed. Further, for a given value of /„ larger mass ratios pro- 
vide higher values of iv The mass ratio of the World War II 
German V-2 rockets was about 3.2. For present rockets R U 
as high as .">. 



Rocket Propulsion 31 



3.9 MULTIPLE-STAGE ROCKETS 



In a single-stage rocket the propulsion energy must be used to 
accelerate the entire empty mass of the rocket even after most of 
that empty mass is no longer useful. This severely limits the 
velocity attainable. Tn fact, with present fuels, a single-stage 
rocket cannot achieve velocities of the order of 25,000 ft/sec and 
higher required to place a satellite in orbit or to escape the earth's 
gravitational field. 

A multiple-stage rocket is made up of a number of independ- 
ent sections each equipped with a propulsion system and a portion 
of the total propellant load. After the first (booster) stage has 
lifted the entire rocket and has reached its burnout velocity, its 
empty mass is dropped from the rocket. A second (sustainer) 
stage carrying the payload is then fired and continues to accel- 
erate the now lightened missile to the appropriate final velocity. 
( )f course more than two stages can be used, but design and oper- 
ational difficulties become more numerous as stages are added. 
If each of a series of stages has the same values of specific impulse, 
dead-weight fraction, payload fraction, and thrust/ weight frac- 
tion, each will contribute the same increase in velocity to the 
final payload. This design results in the lightest over-all rocket 
to perform a given mission. 

A simplified expression for the burnout velocity of a two-stage 
rocket is 

Vt = fvrln (/r,/rs) (3.22) 

Here R t is the initial mass of entire rocket divided by the burnout 
mass of the first stage plus the initial mass of the second stage, 
R = (wioi + v»us)/("'m + »t 02) and R.< m hi,,., ,jj,,,. if the second 
stage is made small in relation to the first stage, the value of the 
logarithmic term in Eq. (3.22) will he greater than that in Eq. 
(3.19), predicting a greater final burnout velocity for the two- 
stage rocket than that given by Eq. (3.19) for a single-stage 
rocket in vertical flight, namely, 1% = i\.rr In R. 



3-10 NUCLEAR PROPULSION 

Some advanced concepts for rocket-propulsion systems have to 
do with development of recoverable boosters, restartable engines, 
s1 "nit>k' propcilants. and nozzles which allow a reduction in 



32 Looking Out: Rockets. Satellites, Space Travel 

engine size. But efforts are also being made to find new sources of 
rocket power, other than chemical reactions;. Figuratively, we 
should like to be able to pack the power output of Hoover Dam 
(1.3 X 10 6 kilowatts) into a sports car. The development of 
nuclear power sources promises to provide specific impulses sig- 
nificantly greater than the values, around 400 sec, for chemical 
fuels. 

Research on the use of a nuclear reactor as a rocket energy 
source has been carried out since 1955 in Project Rover, directed 
by the Atomic Energy Commission and the National Aeronautics 



Pressure shell 



Nozzle 




Reactor core 



Fig. 3.8 Scheme for a nuclear-powered rocket engine. 

and Space Administration. The test engines have been named 
Kiwi's, after a flightless bird. Heat is generated in solid-fuel 
elements by nuclear fission (('hap. 20). Hydrogen gas flows 
through channels in the core. The heated gas is exhausted at high 
speed through a nozzle (Fig- 3.8). 

The thermodynamic (Caniot) efficiency of any heat engine 
is given by 

7', ■ T-> 
Ti 

where 7'i is the temperature (absolute) of the source of energy 
and Ta is the temperature at which the working fluid is dis- 
charged. The lieat -exchanger nuclear engine exhausts into a 
relatively low temperature environment, especially when in 



Efficiency = 



(3.23) 



Rocket Propulsion 33 




Fig. 3.9 KiwI-A nuclear engine at Project Rover test site in Nevada. 



space. So one would expect to be able to put almost all the 
nuclear energy into thrust. The limiting factor is the energy den- 
sity one can put into the propellant to eject it at sufficiently 
high speed. 

\ssumc thai one has an ideal nozzle to recover directed 
kinetic energy from the thermal motion of the propellant mole- 
cules and that the propellant acts as a perfect gas. Then 



W 



RT 



(3.24) 



and the exhaust velocity v c is proportional to \/T/p or to \/T/M, 
where p is the density of the propellant, M is its molecular mass, 
and Ft is the universal gas constant. For high velocities one wants 
maximum temperature and minimum molecular mass. Thus 
hydrogen heated to the highest feasible temperature gives the 
largest specific impulse of any material. Estimates range from 
,J 00 to 1,500 sec for the specific impulse of a heat -ex changer 



34 Looking Out: Rockets, Satellites, Space Travel 

nuclear rocket engine. Recalling the relation n, = 7» p ln (mo/m,), 
it is apparent that the larger l. p attainable with nuclear pro- 
pulsion allows one to reach a desired orbital velocity or escape 
velocity with a much lower initial fully fueled weight (smaller 

mass ratio )«<>/'«»)• 

Under Project Sherwood, studies are being conducted to find 
ways of controlling and using the energy liberated in the fusing of 
the lightest nuclei into heavier nuclei. The phenomena involved 
in thermonuclear (fusion) rockets, plasma rockets, and ion rock- 
ets fall under the general term magn^ohydrodynamica: the study 
of the behavior of ionized gases acted upon by electric and mag- 
netic fields. 

Deuterium is a likely fuel for a fusion rocket. Heated to a very 
high temperature, the deuterium would maintain a high-speed 
plasma (hot, ionized gas) capable of specific impulses rated in 
millions of seconds. There is a difficult problem in confining a 
plasma at the temperatures estimated to he around :i">0 million 
degrees. Perhaps the plasma could be kept from coming into con- 
tact with material walls in chamber and nozzle by suitably shaped 
magnetic fields. 



3.11 ION PROPULSION 

The removal of one or more electrons from molecules of a propel- 
laut, by passing the propelknt through heated metal grids, pro- 
vides ions which can then be accelerated to high velocities through 
a nozzle by an electric field. Volt age takes the place of tempera- 
ture in producing acceleration. 

One such technique uses metallic rubidium or cesium prope.l- 
lant and tungsten grids. Each time an atom of cesium comes in 
contact with the heated tungsten grids, an electron leaves the 
cesium atom and goes to the tungsten metal. The resulting cesium 
ions travel past decreasing potential levels and arc accelerated 
to their final exhaust velocity. 

The ion rocket will always have relatively small thrust. It 
will require assistance (from chemical or nuclear rockets) in 
ground takeolTs where strong gravitational force must be over- 
come. But the performance of an ion engine at high altitudes will 
be very good. Estimates of its specific impulse are as large as 
20,000 sec. The amount of electric power required for an ion 



Rocket Propulsion 35 

rocket is very large. The weight of the electric power plant, even 
using nuclear fission or solar radiation devices, is a major obstacle 
to an efficient ion rocket. 



Electrons 



Distributor plots and 
housing ot 40,000 volts >. 



s El eel 

V-a— i 



Electric generator 
f Electrons 



Propel I ont 




Heater coils ^Ionization gr 

20,000 volts-'' 
10,000 volts- 
5,000 volts 
Fig. 3.10 Scheme for an ion rocket engine. 

Propellent injection 






Arc discharge 




High- current 
circuit 



Fig. 3.11 Scheme for plasma rocket engine. 



3.12 PLASMA PROPULSION 

Hie propcllant may be heated directly by maintaining a powerful 
electric arc in it. In this way high temperatures can be obtained, 
leading to a specific impulse of perhaps 2,000 sec. But this device, 



36 Looking Out : Rockets, Satellites, Space Travel 

too, will require 8 great amount of electric power, about (50 kilo- 
watts for each pound of thrust. 

In plasma and ton propulsion the thrust can be applied con- 
tinuously over an extended period of time. Hence by these tech- 
niques one can propel in space rockets whose weight on earth 
greatly exceeds the thrust of ion propulsion. 



3.13 SOLAR PROPULSION 

In one scheme of solar propulsion, the radiation pressure of solar 
rays (ailing on a "sail," perhaps a lightweight reflecting sphere, 
attached to the spaceship would propel if. In another scheme, the 

Sun's rays heat water 

circulating 

at Focus of mirror, 

producing steam 

Steam-driven 

turbogenerator 

to develop 





Solar sail 



Fig. 3.12 Schemes for solar propulsion, (o) Steam generated by 
solar energy drives electric generator. {b\ Recoiling photons 
impart momentum to sail. 



Rocket Propulsion 37 

solar rays would he used to heat hydrogen gas which would then 
lie expelled through a nozzle. Kstimated values of the thrust are low 
hut are several hundred times those of an ion or a fusion system. 

3.14 MODEL ROCKETRY 

Many a youth has felt the urge to become a backyard rocketeer. 
The National Association of Rocketry, founded in 15)57, seeks to 
advance model rocketry as a scientific hobby and as ati edu- 
cational program. The NAR has developed rules and procedures 
for a safe, supervised, citizen-operated model-rocketry program 
for enthusiasts of all ages. 

Model rocketry is concerned with small, light, inexpensive 
rockets made of paper, balsa, plastic, and other noninetallic 
materials, powered by commercially available rocket motors. 
Emphasis is placed upon design, performance, flight character- 
istics, instrumentation, and reliability. Competitions are spon- 
sored hy local societies. 

I'iiblicat ions ami informal ion about the XAR can lie obtained 
from G. Harry Stine, President, National Association of Hock- 
el rv, Stamford Museum it Nature ( "enter, Stamford. Connecticut. 

Physics teachers may he interested in model rocketry as a 
device for stimulating student interest in mathematics, mechan- 
ics, aerodynamics, meteorology, electronics, optics, and pho- 
tography. Ideas based on the experience of the most active sections 
of the NAIl may be requested from Dr. Stine at the address 
above. 



MANUFACTURERS OF MODEL ROCKETS 

American Telasco Limited, 135 New York Ave., IhUesitc, X.V. 
Centuri Knpncering Co.. 340 \V. Wilshirc Drive, Phoenix 3, Ariz. 
Ci>a»ler Corporation, P.O. Box 2S0, Hiiless. Tex. 
Bates Industries, Inc.. P.O. Box 227. Penrose, Colo. 
Model .Missiles, Inc., 2ti!<) Bast Cedar Ave., Denver 22, Colo. 
Propulsion Dynamics. Inc., P.O. Pox 2XXA. Ut. 1, Officii, Utah 
Rocket Development Corp., Box 522, Rich mood, bid. 
Cnited Scientific Co., Inc.. P.O. Box S9, Waupaca, Wis. 

FILM 

The Itislor,/ ami Development of the fiwkri. 10 min (1962). MeUruw-llill 
Text-Film Division, 330 West 42 St., New York 36, N.Y. Available 
in color or black and white. 



38 Looking Out: Rockets, Satellites, Space Travel 



SUGGESTIONS FOR FURTHER READING 

Hobbs. Marvin: "Fundamentals of Rockets, Missiles, and Spacecraft," 
John F. Rider, Publisher, Inc. New York, L962, 27ft pp. 

"An Introduction to Rocket Missile Propulsion," Rocketdync, Canoga 
Park, Calif.. 1958, 12ft pp, 

"Model Kits," Revcll, Inc., 4223 Cileneoe, Venice, Calif. 

"The Next Ten Years in Space, 1959-1989," House Document 115, 
U.S. Government Printing Office, Washington 2ft, DC., 1959, 

221 pp. , 

"1959 Missiles and Rockets Encyclopedia," Re veil, Inc., Venice, 

Calif., 32 pp. 

"Physical Data, Constants and Conversion Factors," General Electric 
Missile and Space Vehicle Department (Mail: P.O. Box Sftfto, 
Philadelphia 1, Pa.), 1959, 24 pp. 

•■Rocket Experiment Safety: Safety Suggestions for the Rocket Hobby- 
ist/' Atlantic Research Corp., Alexandria, Va., 1958, 19 pp. 

Seiferl, Howard S.. Mark M. Mills, and Martin Summerlield: The 
Physics of Rockets, American Journal of Physics, 15:1-21, 121-140, 
255-272 (1947). 

"Space Facts: A Handbook of Basic and Advanced Space Might and 
Environmental Data for Scientists and Engineers," General Elec- 
tric Mis>ile and Space Vehicle Department, Valley Forge Space 
Technology Center (Mail: P.O. Box 8555, Philadelphia 1, Pa.), 
I9f>0. fil pp. 



QUESTIONS AND PROBLEMS 

1. Verify the statement: "Near the surface of the earth, gravity 
robs a vertically rising rocket of about 20 mi/hr in speed each second, 
or about 2,400 mi/hr for each 2 iniu of acceleration." ^ 

2. A projectile is Srcd with a speed of 300 ft/sec at an angle ot 87 
with the horizontal. Compute the speed when it first reaches a height of 
.jqq- fj._ j4«s. 83.5 ft, sec 

3. A force of 4,900 dynes acts on a 20-gm mass for 8.0 sec. (a) What 
acceleration is caused? (I>) Bow Tar docs the mass move from rest in the 
8.0 sec? («) How fast is it going at the end of S.O see? 

Am. 24ft em/see, 8,140 cm, 1,900 era/see 

4. Show that the mass ratio tin/m,, for a multistage rocket is the 
product of the mass ratios of its individual stages. 

5. Would you consider an alkali metal, such as cesium, a prospec- 
tive propellant for an ion rocket? Why? 

6. If it becomes possible to convert nuclear-fusion energy of a plasma 
directiv into electric energy, without the conventional rotating gener- 
ator, would this make ion propulsion of rockets more feasible? 



Rocket Propulsion 39 

7. Show that if one increases the exhaust temperature of a hc-al- 
cNclianger nuclear rocket, the specific impulse and power requirements 
will increase as J' ! and the mass ratio will decrease as exp T. What 
limits this favorable picture? 

8. The Atlas 1CBM is called a I '-stage rocket because of its 
unique application of the step principle. The Atlas has three main 
engines: two booster engines and one suslainer. Each engine receives 
propellant from a single very lightweight tank. The three engines are 
mounted parallel to one another. All three engines are ignited at take- 
ofT. I/titer. at staging, the boosters ami housing slide backward on rails 
and drop to earth, leaving the suslainer engine to propel (he vehicle. 
Can you suggest some advantages of this type of staging over the con- 
vent iona! tandem staging? 

9. How can rocket action be demonstrated with a toy balloon? 

10. What is the fallacy in the following argument? "A horse pulls 
on u cart. By Xewton's third law, the cart pulls back on the horse with 
a force etjiuil to that exerted by the horse on the earl. Hence the sum of 
the forces is zero, from which it follows that it is not possible for the 
horse to accelerate the cart." 

11. Comment on the remark, "Space stations will be obsolete when 
they are feasible." 

12. A rocket whose thrust is 27,000 lb weighs initially 22,000 lb, of 
which SO per cent is fuel. Assuming constant thrust, find the initial 
acceleration and the acceleration just before burnout. Xeglect air 
resistance and variation of g. Arts. 39.2 ft/sec 2 , 190 ft/sec 2 

13. From Eq. (3.19) show that for a rocket launched horizontally 
and continuing in a path parallel to the earth, the burnout velocity is 
given by Vt = err In R. 

II. What is the minimum value of mass ratio R for which the 
burnout velocity n, of a rocket will exceed the effective exhaust ve- 
locity iv„? 4 ns. R > 2.718 

15. Do you agree with Professor Fink's comment that Hie methods 
of achieving lift listed in order of increasing sophistication of the under- 
lying physical principle are (1) satellite vehicles, (2) displacement of 
lighter-than-air craft, (3) hover craft or ground-cushion vehicles, 
(4) vertical flight rockets, (5) vertical takeoff and landing machines, 
(<>) conventional airplanes? If so, how do you account for the historical 
fact that the "simplest" methods were not the first to be suecessfullv 
Used? 



Escape 
from Earth 



No thing is too high for the daring of 
mortals : We storm heaven itself in our 
folly. Horace 

Every great advance in science has 
issued from a new audacity of imagi- 
nation. John Dewey 



Through the ages men have dreamed of the power of flight. In 
Creek mythology Daedalus and Icarus made a daring ascent 
into the air on wings made of birds' feathers and wax. In the 
notebooks of Leonardo da Vinci are found detailed drawings of a 
flying machine. With the success of the Wright brothers, man 
began to realize his long ambition of flight through the air. But 
now he turns his dreams to flight beyond the enveloping and 
protective atmosphere — into space. 

In designing a vehicle to escape the earth, one has to solve the 
problem of piercing the earth's atmosphere. A second, more 
troublesome, problem of escape is that of overcoming the force 
of gravity. Since each body in the universe has its own gravita- 

40 



Escape from Earth 41 

tioual field, a vehicle in space would encounter an endless mixture 
of gravitational fields, one superposed on another. The terms 
escape and capture refer to the transfer of the vehicle from one 
field to another. 

4.1 GRAVITY 

In addition to the three laws of motion, Newton formulated the 
law of universal gravitation: Every particle in the universe 
attracts every other particle with a force that is directly propor- 
tional to the product of the masses of the two particles and in- 
versely proportional to the square of the distance between their 
centers of mass 



F = G 






(4.1) 



where F = force of attraction 

m [ and mi = masses of the two particles 

s = distance between them 

G = gravitational constant, whose value depends on 
the system of units used 
In mks units 

G = (5.07 X 10 " newton-mVkg s 

Gravity acts as a brake on a vehicle leaving the earth. While 
traveling in space, a vehicle is always subject to some gravity. 
The vehicle attracts and is attracted by all celestial bodies. But 
because gravitational force follows an inverse square law (/•' <* 
1 «-), the mutual attractions of only the nearest bodies are usually 
significant. When a vehicle returns to earth, it is accelerated by 
an increasing gravitational force. 



4.2 FREELY FALLING BODIES 

An unsupported body starting from rest near the surface of the 
earth drops 10 ft during the first second, 04 ft at the end of the 
iH'Xt second, 144 at the end of the third, etc. It has an acceleration 
r, f "V2 ft per sec per sec, or 32 ft/sec-. The symbol g is used to 
represent the acceleration due to gravity. At sea level and 4:">° 
latitude, g has a value of 32.17 ft/sec 2 or 9.806 m/sec 2 . 

The value of g varies slightly over the earth owing to local 



42 Looking Out: Rockets, Satellites, Space Travel 

variations in mass distributions and to the fact that the earth 
bulges slightly ai the equator. Surface gravity values vwy from 

planet to planet owing to differences in mass, radius, and rota- 
tional speed. 

When air resistance can be neglected, the equations in Sec. 3.2 
for uniformly accelerated motion apply to falling bodies. 



t, sec 


1 



v, ft/sec s, ft 



O 
O 



°\ 




32 



64 



96 



128 




16 

64 



144 



256 



Fig, 4.1 Position and speed of a body 
falling freely from rest after successive equal 
time intervals. 



4.3 GRAVITATIONAL FIELDS 

The force which one body exerts on another at a distance is con- 
veniently described by the "force field" set up by one of the 
bodies, the source. Various kinds of forces can be treated in this 
way. Electric charges exert forces upon other electric charges. 
Magnets exert forces on other magnets. Matter exerts gravita- 
tional force upon other matter. 

The force exerted on a unit test particle (unit charge, unit 
mass, etc.) has a definite magnitude and direction for each pos- 
sible location of the test particle. The whole assemblage of these 
for08 vectors, or the mathematical function relating force to 
position, is called a field of force. Any path that would be fol- 
lowed by a free incrtialess test particle is called a line of force. 

The gravitational field intensity / at any point A in the 
space near a mass ,1/ is defined as the force per unit mass acting 
on any mass m placed at A : 



/ = 



m 



(4.2) 



Escape from Earth 43 

The small mass m is used only as a means of detecting and measur- 
ing the gravitation field, Whether m is large or small, the. force 
per unit mass placed at .4 has a definite value, J. 

By substituting Eq. (1.1) for F in Kq. (4.2), we get the 
expression for the gravitational field intensity / at a distance r 
from niitss .1/ in terms of the universal gravitational constant Gas 



m r 2 



(4.3) 



When one knows the field intensity, one can find the force acting 
on any mass m as the product of m and /, 




F 



Fig, 4.2 Gravitational forces of attraction. 




Fig. 4.3 Parabolic poth of a projectile in a uniform gravitational 

field. 

For a freely falling body, Newton's second law of motion 
becomes F = mg. At the earth's surface, therefore, the gravita- 
tional field intensity is equal to g, the acceleration due to gravity 



' = - = ? 
m 

I newton 



(4.4) 



meter 



kilogram second 2 

111 mka units, g = 9.80 m/sec 2 



44 Looking Out: Rockets, Satellites, Space Travel 

In a region of free space where the gravitational field is prac- 
tically constant in direction and magnitude, the path taken by a 
projected mass m is a parabola (Kig. 4.3). 

4.4 GRAVITATIONAL POTENTIAL ENERGY 

To find the work needed to get off the earth, let us calculate the 
work done in moving a mass m from the surface of the earth, 
radius R, to a distance r from the center of the earth. Imagine 



Distance 
r * measured 
_ from C 



Fig. AA Calculation of gravitational poten- 
tial energy. 




the distance from 5 tor to be divided into small equal intervals so 
that over each the gravitational force F a will lie practically con- 
stant. Then we can easily calculate thr work done in each interval 
and add to get the total. At the surface, Fa = QMjm A' 1 '. At the 
top of the first interval Fa is OMjm r,«. Since these values are 
nearly the same, we can use for the average force in the first 
interval (!M<w ltr u The work done in the first interval is then 

Wt = Fair, ~ 8) = Vg fir. - *> = W* (l " ,') 

Likewise the work in the second interval is 

„,. 9£ h -„-«.„.„, (i-I) 



Escape from Earth 45 
and in the third 

Wt = GMjn (- - -) 

If we add these three expressions, the intermediate values r, and 
n cancel out. The work done in the first three intervals can be 
expressed in terms of the values of r at the ends: It and r 3 . Thus 



W = GM 



- m (it ~ ' ) 



(4.5) 



is the general expression for the work required to move a mass in 
against the earth's gravitational field out to a distance r. By 
definition this is the gravitational potential energy of mass m in 
the field of the earth. 



4.5 VELOCITY OF ESCAPE, FROM CONSIDERATION 
OF ENERGY 

To estimate the maximum height attained by a rocket fired 
straight up, we may equate its kinetic energy at burnout to the 
gravitational potential energy it acquires thereafter in rising, 
with decreasing speed, to its maximum height, t x 



^mv 2 = mgR* 



\lt ''i..:. </ 



(4.0) 



Example. A roirkci has an upward speed uf 5.0 mi/see :it burnout. Find 
the maximum height it attains. 
From Eq. (4.6) 

mf-n mi Y m ft (*ixm ■ 3960- mi' \ 
V 880/ "»* \ r.„n< ) 

25 / miV / 32 \ .. i(irin { iniV ( 82 \ /39(i0= mi A 

%\&) =Uoj c ^ o) tcj -U»/U.*W 

r n , ax = 8,S(W mi 
(What amplifying assumptions have been made in this Bohition?) 



4 -6 VELOCITY OF ESCAPE, FROM CONSIDERATION OF FORCE 

If a gun on a cliff overlooking the ocean fires a bullet horizontally, 
the bullet will strike the water at some distance from the base of 
■he cliff. If the initial speed of the bullet is increased, the range 



46 Looking Out : Rockets, Satellites, Space Travel 

is increased. For a particular speed, which depends on the dis- 
tance of the gun from the center of the earth, the hullet would 
make a complete circuit of the earth, at a constant altitude, A 
(Fig. 4,">), If it- did not encounter resistance, it would continue 
to move in orbit about the earth. 

Prom Newton's second law, F = ma, and Eq. (4.2), the 
force needed to hold the bullet in a circular path at altitude A is 




Fig. 4.5 Range increase* with horizontal firing speed until circular (orbital 
speed is reached. 



inv-fQi + R). This force is provided by the gravitational attrac- 
tion of the earth, so 



/' - 



CM ,i» i-ni 

(/,' + /,)■' H + h 



(4.7) 



Example. At what speed would a projectile have to leave a platform, 
horizontally, 300 mi above the earth in order to enter a state of "con- 
tinuous fall" around the earth? 
From Kq. (4.7), 

C.U e 6.67 X IP"" newton-m* 5.983 X 10" kg 



ys = - 

R + h kg' 

= 58.1 X ICm'/sec* 
w = 7,«2() in sec = 1,700 mi/hr 



mi 



(3,950 + 300) mi 1,609 m 



4.7 WEIGHTLESSNESS 



A body in orbit around the earth or following an unrestricted, 
un powered course in a gravitational field anywhere in space is 
said to be in "free fall," also called a state of "zero gravity." 
Actually, gravity is not absent. The force of gravity continually 



Escape from Earth 47 

acts on the body and determines its path. But the condition of 
weightlessness is experienced because there is nothing to resist 
the body's motion in response to gravity. 

Human beings have experienced weightlessness for the first 
few seconds after leaving a high diving board, or for somewhat 
longer periods in aircraft on "zero g" trajectories, and more 
recently in manned rocket flights. 

4.8 POTENTIAL-WELL MODEL 

Using Eq. (4."»), we may plot a graph showing the potential 
energy K p which a body of mass ni would have at various dis- 
tances r from the center of the earth (Kig. 4.6). When the mass m 



50,000 



25.000 



,,„, ennnn ■ S GrOVltOtlOnol 

25,000 50, 000 mi f , 
1 1 £■ free •■ 




: spoc 



Fig. 4.6 Gravitational potential energy of mass m, showing "well" analogy for 
earth's field, 



is infinitely far from earth, li p = 0. As mass m is brought closer 
and closer to the earth, work is done on m by the earth's field and 
the potential energy of m acquires a larger and larger negative 
value. Thus on the surface of the earth we live in a gravitational 
well thousands of miles deep. To reach the moon or another 
planet we must climb out of this well onto the plane marked 
"gravitational free space" in Fig, 4.(i. 

A potcntial-energy-well model for demonstrating satellite 
orbits may be made from a suitably shaped wine glass to reprc- 



48 Looking Out: Rockets, Satellites, Space Travel 

sent the surface (Fig. 4.7) obtained by rotating the graph of Fig. 
4.6 ahout its vertical axis. A marble representing the satellite 
may be caused to travel a variety of orbits by varying its initial 
velocity. 




Fig. 4,7 Potential -well model for demonstrating circular 
(c) and elliptic (e) orbits. ({See J. S. Schooley, Satellite 
Orbit Simulator, American Journal of Physks, 30: 531-532 
(1962).] 



QUESTIONS AND PROBLEMS 

1. What is the largest gravitational force of attraction between 
two solid metal spheres cadi of 50.0 kg mass and 10.0 em radius? How 

does lllis force compare with the force of attraction of the earth on 
each sphere? 

Ans. 4.17 X 10"° newton, weight is 120 million times larger 

2. What would be the value of <j. the acceleration clue to gravity, 
if the earth had half its present diameter? 

3. If the mass of the moon were doubled hut the orbit remained 
the same, what would he the period of the moon? 

t. A 100-lb man starts sliding down a rope with a downward 
acceleration of p/S. (a) What is his apparent weight? (b) What is the 
tension in the rope above the man? 

5. Using the experimentally determined value of (7 and the distance 
93 X 10° mi from carlh to sun. calculate the mas* of the sun. 

6. At what point in its trajectory does a projectile have its mini- 
mum speed? 

7. If a rocket at tains a speed of (500 mi hr hy the time it reaches 
1,000 ft, how many times g is its acceleration? 

8. The earth revolves about the sun in a nearlv circular orbit 



Escape from Earth 49 

(r = 150 X 10 e km) with a speed of about 30 km see. What is the 
acceleration of the earth toward the sun? 

9. Show that to escape from the atmosphere of a planet, a molecule 
of gas must have a speed r such that t ! > 2C.i//r, where .1/ is the mass of 
the planet and r is the distance of the molecule from the center of the 
planet. What hearing does this have on the composition of (lie atmos- 
phere surrounding the earth and other planets? 

10. A balloon which is ascending at the rate 12 m/sec is 80 m above 
the ground when a -lour is dropped from it. How long a time will be 
required for the stone to reach the ground? Ans. 5.4 sec 

11. An elevator is ascending with an acceleration of 4.0 It sec 2 . At 
the instant its upward speed is K.O Ft/see, a holt drops from the top of 
the cage 9.0 ft from its floor. Find the time until the holt strikes the 
floor and the distance it has fallen. Ans. 0.71 sec, 2.3 ft 

12. A body hangs from a spring balance supported from the roof 
of an elevator, (a) If the elevator has an upward acceleration of 4.0 ft/ 
sir- and the balance reads 45 |b, what is the true weight of the body? 
(ft) In what circumstances will the balance read 35 lb? {<■} What will the 
balance read if the elevator cable breaks? 

Ans. 40 lb, a = 4.0 ft/sec s downward, zero 

13. If the mass of the moon is ^ the mass of the earth and its 
diameter is -J- that of the earth, what is the acceleration due to gravity 
on the moon? How far will a 2.0 gm mass fall in 1.0 .-cc on the moon? 

Ans. I g, 3.2 ft 

14. A girl standing on a diving hoard throws a ball with a hori- 
zontal velocity of 50 ft/sec to a man in the water. In doing so, she loses 
her balance, falls off the hoard, and strikes the water in 2.0 sec, (a) How 
far is the man from the base of the diving board? (b) How high is the 
diving hoard above the water? (e) What is the velocity of the ball at 
the end of its path? 

.bis. 100 ft, 64 ft, SI ft/scc, at 52° with the horizontal 



Satellites 51 



Satellites 



It is no paradox to say that in our most 

theoretical moods we may be nearest 

to our most practical applications, 

A. N. Whitehead 



The launching of the first artificial earth satellites, the Russian 
Sputniks I and II, in 1957, aroused worldwide interest in the 
power and control attainable with rockets. The special scientific 
investigations made during the International Geophysical Year 
were significantly aided by data from instruments carried in 
satellites. From the orbit of a satellite one may better estimate the 
shape and dimensions of the earth. A permanent satellite can be 
useful as an aid in the navigation of ships, aircraft, and missiles. 
For a satellite which eventually returns to earth, measurements 
of the orbit may yield a more precise value of g. Atmospheric 
drag and the effectiveness of radio emission at various altitudes 
can be studied. Equipped with suitable instruments, a satellite 
can also measure solar and cosmic radiation, temperature and 
pressure variations, and the distribution of the earth's magnetic 
field. In short, satellites can tell us much that we want to know 
about our earth and much that we need know about space hazards 
before we venture into space ourselves. 

50 



5.1 ELEMENTS OF AN ORBIT 

To define the position of an earth satellite in the solar system and 
to describe its path, one needs to know the period of the satellite 
and the elements of its orbit, that is, the constants which fix its 
position and shape in space: 

The period is the time for a satellite to make one revolution 
around the earth. 

The perigee is the position of closest approach to the center 
of the earth. Apogee is the position of the satellite farthest from 
the earth (Fig. 5.1a), 

The eccentricity describes the flatness of the orbit as the ratio 
of e to a (Fig. 5.1/j). Here e is the distance from the center of the 



Apogee 




Perigee 




Fig. 5.1 Elliptical orbit. 



orbit to the focus at the center of the earth, while a is the semi- 
major axis. 

The angle of inclination i of the orbit is the angle between the 
plane of the orbit and the plane through the equator (Fig. 5.2). 

The plane of the satellite orbit intersects the equator plane in 
a straight line called the line of nodes. This line intersects the 
satellite orbit at two points, called nodes. At one of these, the 
ascending node, the satellite crosses northward from "below" 
the equator plane to "above" the equator plane. At the other, 
the descending node, the satellite crosses southward from "above" 
th f equator plane to "below" the equator plane. The orbit ele- 
ment we now define is the longitude V, of the node, or the angle of 
"■•<>•' tiding node. This angle, P. in Fig. ,5.2, is measured in the plane 
J f the equator from the direction of the vernal equinox to the 
unction of the ascending node. (To describe the motion of a 




-Q 

o 






Satellites 53 

planet about the sun, one substitutes "ecliptic" for "equator" 
in the definitions above.) 

The argument of perigee us is the angle measured in the orbit 
plane between the direction of the ascending node and the direc- 
tion of the perigee. 

To summarize, the elements of an orbit are period, perigee 
apogee, eccentricity, angle of inclination, angle of ascending node, 
ami argument of perigee. 



5.2 CIRCULAR ORBIT 

The path of any body acted on only by an inverse square force 
(/-' a l/r ! ) due to a neighboring fixed body will ho an ellipse, 
circle, parabola, or hyperliola (Chap. (i). To simplify our analysis, 
for the remainder of this chapter we shall examine an earth 
satellite in a circular orbit and consider only the interaction be- 
tween the earth and the satellite. Although small perturbations 
may be produced by the atmosphere, the moon, other planets, 
and satellites, for the present these effects will be neglected. 

For a satellite in circular orbit, the gravitational force exerted 
on it by the earth has no component in the direction of motion 
which could either increase or decrease the speed of the satellite. 
It orbits at constant speed. 

The force on the satellite is given by Xewton's law of 
gravitation 



F = G 



Mm 



(5.1) 



where the mass of the earth M and the mass of the satellite m are 
icffiirded as concentrated at the center of each, a distance r apart. 
The constant of gravitation G can In* determined in the labora- 
tory. Because the mass of the earth is so very large, the center of 
mass of the two bodies is practically at the center of mass of the 
earth. The motion may be described as a circular motion of the 
satellite about a fixed center of force. 

The direction of the velocity of the satellite in circular orbit 
» continually changing (Fig. 5.3). Gravitational force continu- 
ally produces a "centripetal" acceleration a toward the center. 



54 Looking Out: Rockets, Satellites, Space Travel 



Satellites 55 



For this uniform circular motion the acceleration is v-/r and the 
centripetal force is mv*/r. 



/ y — -\A 




v ¥ canst. 

\13\- const. 
\a\— const. 



Fig, 5.3 A satellite in circular orbit is continually 
accelerated toward center of orbit. 



5.3 PERIOD 

We may equate the gravitational force and the centripetal force 



f ,Mm _ mv' 1 
G — -z — ~rr 



(5.2) 



From this equality we find the speed that a satellite has to obtain 
to maintain a particular altitude 



GM 

v = x - 



(5.3) 



The mass m of the satellite does not appear in the equation for 
speed. The closer the satellite is to the earth, the greater must be 
the speed, because the gravitational attraction is greater. 
Since the angular speed tn is 2w/ "period, 



2x JOM 



and since 

we obtain an equation. 



7'2 = 



GM 



(5.4) 



which says that the square of the period of the satellite is propor- 
tional to the cube of its distance from the center of the earth. 
This is Kepler's third law of motion, for the special case of a 
circular orbit. 



5.4 ENERGY 



The total energy remains constant in satellite motion. This can 
be shown very easily for our special case of a circular orbit. 
Substitution for the speed of the satellite (Eq. 5.3) into the equa- 
tion for kinetic energy E k gives 



v i— * GMm 
h k = \mv* = - 2r 



(5.5) 



Since the gravitational potential energy Ey of the satellite is 
GMm 



B, = - 



1 1 ie total energy is 

E k + E„ «■ - 



GMm 
2r 



(5.6) 



(5.7) 



The total energy is negative for both circular and elliptical orbits. 
This means that the satellite is bound to the center of force and 
cannot escape unless sufficient positive energy is provided (see 
*'ig. 4.7). 



5.5 ANGULAR MOMENTUM 

Iho total angular momentum L of a satellite moving at constant 
speed in a circular orbit is the product of its linear momentum mv 



56 Looking Out; Rockets, Satellites, Space Travel 

and the radius r. Vcctorially, the angular momentum is repre- 
sented by a vector L drawn to scale to represent the scalar mag- 
nitude mvr and drawn along a line perpendicular to the plane of 
r and v in the direction indicated by tlie thumb of the right hand 
when the fingers are allowed to curt from the direction of r into 
the direction of v. Thus L results from a vector "cross product," 
the notation for winch is 

L = rX (hit) 

It is obvious that for a satellite in uniform circular motion 
the total angular momentum is constant. This is also true when a 



*S^ 




Fig. 5.4 Angular momentum of mass m is re- 
presented by vector I. 



satellite moves in an elliptical orbit. The radius and speed vary, 
but the total angular momentum remains constant. This is 
equivalent to Kepler's second law, that a line joining the focus 
and the satellite sweeps out equal areas in equal periods of time 
(Fig. 5.5). 




Fig. 5.5 The satellite sweeps out equal 
areas in inequal periods of time. 



It is not feasible or even particularly desirable to launch a 
satellite into a perfectly circular orbit. If such an orbit were 
attained, slight perturbations would soon make it elliptical. 

Observation of a satellite in orbit gives us information about 
irregularities in the shape of the earth. As the satellite orbits, the 
plane of its orbit rotates or regresses toward the west. At the 
same time the orbit turns in its own plane, swinging the perigee 
around. Also, both ends of the orbit become somewhat flattened. 



Satellites 



57 



These observations are interpreted as proof that the earth 
bulges slightly around the equator, owing to the earth's rotation. 
The gravitational force tends to pull the satellite toward the 
equator. Consider the gyroscopic property of the satellite. The 
gravitational force due to the bulge tends to tip the axis of the 




Earth's rotation 



Fig. 5.6 Rotation of the earth and precession of the satellite orbit expose 
different areas of the earth to the satellite, os shown in Fig. 5,7. 



orbit. The reaction causes the plane of the orbit to prccesK around 
the earth in a westerly direct ion, while the earth is rotating from 
"est to east. This precession may be an advantage in the case of 
Certain types of observational satellites which thus may "see" 
most of the earth's surface (Kigs. 5.6 and 5.7). 




Satellites 59 



QUESTIONS 

I. Why is tin* upper (dotted) path in i)k> accompanying sketch not. 
a possible satellite orbit about the earth? 



Fig. 5.8 




2. Show ilial if frictional forms cause ;i satellite to lose total energy, 

it will move into an orbit closer to (he earth with an actual increase 
in speed. 

3. After a certain satellite was put in orbit, it was stated thai the 
satellite would not return to earth but would burn up on its descent. 
I low is this possible, since it did not burn up on ascent? 



Motion of Bodies in Space 61 



Motion of Bodies 
in Space 



If I have seen farther than ethers, it is hy 
standing on the shoulders of giants. An 
old saying quoted by Newton 



A space vehicle when not under power is governed by the same 
laws which determine the motions of stars, planets, and comets. 
These laws are Newton's law of universal gravitation and Kep- 
ler's taws of planetary motion. Karly in the seventeenth century, 
Kepler by inductive reasoning formulated his three laws to fit 
the astronomical observations and calculations made available 
1 1! him by his patron Tycho Brahc. Xewton in his "Principia 
Mathematica" (Ih'87) showed that the kind of planetary motion 
described by Kepler's laws can be deduced from the universal 
law of gravitation. 

6.1 KEPLER'S LAWS 

Kepler's description of planetary motion may be stated as follows; 

Law I. The planets move in ellipses having a common focus 

situated at the sun. 



Law II. The line joining the sun and a planet sweeps out equal 

areas in equal periods of time. 
Law III. The square of the period of a planet is proportional 

to the cube of its mean distance from the sun. 

An ellipse may be constructed by using two pins and a loop 
of string to guide a pencil (Kig, 0. 1). This method of construction 



Fig. 6.1 
ellipse. 



Construction of an 




makes use of a geometrical property of the ellipse: The sum of 
the distances from any point on an ellipse to the two foci, A and 
li. is constant. An ellipse with its center at the origin of coordi- 
nates and with foci on the x axis is represented by an equation 
<>!' the form 

+ £- = 1 
a 2 T b* 

From Kepler's second law, if the shaded areas in Kig. (>,2 arc 

S3 

x, f~~^ 

Fig, 6.2 Law of areas. 




a " equal, a planet takes equal time intervals to travel the dis- 
tances St, $ 2l and s a . 

Kepler's third law, called the harmonic law, expresses the 
Proportionality of period squared, 7", and the cube of the scmi- 
Boajor axis a of the ellipse. 



60 



62 Looking Out: Rockets, Satellites, Space Travel 

Example. Calculate the height of a satellite in a 24-hr orbit about the 
earth if it has been observed that a satellite at a mean distance of 
4,100 mi from the center of the earth has a period of 5,000 sec. 
From Kepler's third law 



3Y 
7V 



a.* 



we wish to find o 2 when T~ = 1 day = 8.6 X 10 4 see 
= (I/V a , = [tf)l 4,100 mi = 27,000 mi 



a-i 



Kepler's lawn apply to the ideal ease of only two bodies mov- 
ing under their mutual gravitational attraction. But in space 
travel, effects of other bodies have to he considered. 

To consider the feasibility of certain proposals or devices, 
one starts by examining qualitative orbits. Such trajectories are 
predicted with the aid of simplifying assumptions: that the moon 
moves in a circle around the earth, that the earth may be con- 
sidered symmetrical, that any disturbing masses are in the orbit 
plane of the space vehicle, etc. The precision trajectories needed 
for actual space travel do not allow these approximations. Hence 
the calculations become enormously more complicated.* 



6.2 NEWTON'S DERIVATION OF KEPLER'S LAWS 

As a test of his theory of universal gravitation, Newton desired 
to show that Kepler's laws could be derived from the law of gravi- 
tation and he desired to investigate the more general problem: 
What kind of motion is necessary according to that law? In its 
basic statement, the law of universal gravitation applies only to 
particles ("point" masses). Newton needed first to show that the 
attraction for an exterior particle exerted by a spherical mass 
(either homogeneous or somewhat like the earth, made up of 
concentric homogeneous shells) was directly proportional to the 
total mass of the sphere and inversely proportional to the square 
of the distance of the particle from the sphere's center. Newton's 
difficulty in establishing this principle to his satisfaction may have 
been the cause of his delaying some twenty years in publishing 
his conclusions. 

* Precision rocket orbits are discussed in S. Herrick, "Astrody- 
namies," I). Van Xostrand Company, Inc., Princeton, N'..l., 1959. 



Motion of Bodies in Space 63 

The orbits of all the planets (except Pluto) are very nearly 
circles, with the sun at the common center. Kepler's third law 
can be derived by equating the centripetal force to the gravita- 
tional force (Sec. u.3) to obtain 

T* 4» ! 

i* = m = con8tant 

Kepler's second law, the law of equal areas, follows whenever 
the interaction between two particles is in the direction of the 
line joining them. The force need not follow an inverse square. 




Fig. 6.3 Derivation of Kepler's second law. 

Let Pi, P->, and P% be points along a planet's orbit marking the 
position of the planet at time intervals of 1 sec. Then the distance 
PiPz is numerically equal to the planet's velocity 1% and /V J 3 is 
numerically the velocity r., in the next second. When the only 
force acting on the planet is in the direction of the sun, this force 
has a component zero perpendicular to line / J 2 ,S'. Hence the com- 
ponent of the planet's velocity perpendicular to / J -,.S' must be 
unchanged, according to Newton's first law of motion: r tL = u 2l . 
The area swept during the first second by the line joining the 
planet and the sun is &P1P1. The area swept in the next second 
is SP t P 3 . These triangles have the same base P°S and equal alti- 
tudes v ± ; hence they have equal areas. 

The derivation of Kepler's first law is more lengthy, and it 
involves differential equations.* The question is: Given an in- 
verse-square law of attraction, what shape must a planet's (or 
comet's) orbit have? The answer turns out to be: The orbit will 
be one of the conic sections with the attracting body (sun) in 
one focus, 

* A derivation without calculus is presented in Jay Orear, "Funda- 
mental Physics," pp. 70-73, .John Wiley & Sons, Inc., New York, 19GI. 



64 Looking Out: Rockets. Satellites, Space Travel 

Conic sections are curves? obtained by taking plane slicas of a 
solid circular cone (Fig. G.4). The cone sliced parallel to its base 
(I) gives a circle. If the cut is slanted, the section is an ellipse (2). 
With greater slant, the section is a parabola (3). With still greater 
slant, the section is a hyperbola (4). 





1. Circle 

2. Ellipse 

3. Parabola (parallel to lineaO) 

4. Hyperbola 



Fig. 6.4 Basic orbits related to conic sections. 




Fig. 6.5 Newton's proposal for an earth satellite, [(a) From Sir Isaac Newton, 
"Mafnemoficai Principles . . . ," edited by F. Co/oW, University of California 
Press, Berkeley, Calif., 1934. (b) From E. M. Rogers, "Physics for the inquiring 
Mind," Princeton University Press, Princeton, N./., 1960.1 



Motion of Bodies in Space 65 

The significance of the various conic-section trajectories may 
be clarified by an example based on Newton's own suggestion for 
an earth satellite. About 1660 he predicted in a drawing (Fig, 
(5.o) that if a cannon ball could be fired with a muzzle velocity of "i 
mi/sec, it would circle the earth as shown. The Sputnik and 
Explorer satellites did achieve this velocity, For a low-flying 
earth satellite in a nearly circular orbit, equating the centripetal 
acceleration r/r to g gives 

v = \/gr — 9.8 X (>.' r > X 10 6 m/sec = 8 km/sec or 5 mi /sec 

Now alter Newton's drawing (Fig. (U>) by considering the 
mass of the earth to be concentrated at point E (Fig. (i.(i). Con- 



Hyperbola 




Fig. 6,6 Conic orbits cotangent at satellite launching point p. 

^ider that a satellite is to be launched at point p with a velocity 
Perpendicular to the line K v . J-et the, circle represent the orbit of 
the satellite described in the preceding paragraph. The effect of the 
earth's attraction is to cause the satellite to fall r = Igl 1 = 4.9 m 
toward the earth in the same second it travels 8 km along the 
tangent. The two displacements bring the satellite back to the 



66 Looking Out: Rockets, Satellites, Space Travel 

same distance it had before. So, during each second, the satellite 
falls toward the earth but never gets any closer. 

Now suppose that the satellite's velocity is made less than 
S km/sec. The earth's effect of 4.0 m each second is unchanged. 
So the satellite will fall closer to the earth along the smaller 
ellipse of Pig. 6.0. Since the earth is not a "point" as implied in 
Fig. 0.0, the satellite actually will not he able to complete the 
elliptical orbit but rather will strike the earth after traveling a 
trajectory which is si portion of an ellipse (Fig. 6.5). (The smaller 
ellipse of Fig. 0.0 could represent the path of a comet or planet 
about the sun at A'.) 

If the satellite at p were given a velocity somewhat greater 
than 8 km/sec, the 4.9 m by which it would fall to the earth each 
second would be insufficient to hold the distance constant. The 
satellite would climb away from the earth on the larger of the two 
ellipses (Fig. 0.0). With decreasing speed the satellite would 
arrive at a point a opposite the start. There the centrifugal 
reaction would be insufficient to overcome gravitational attrac- 
tion, though the latter would also have decreased. Accordingly, 
the satellite would begin to fall back toward the earth, regaining 
speed along the elliptic path until it reached point j> with the same 
velocity as at the start. 

Increasing the satellite's velocity at p still more would semi 
it off along the parabola shown in Fig. 6.6. Still greater velocities 
would carry the satellite away from the earth along a hyperbolic 
path. In either case the attraction of the earth would be insuffi- 
cient to decrease the radial velocity of the satellite enough to 
cause it to return. 



QUESTIONS AND PROBLEMS 

1. The periods of revolution of the planets Mercury, Venus, Mars, 
and Jupiter are, respectively. 0.241, 0.017, 1.88, and 11.9 years. Find 
Iheir mean distances from the sins, expressed in astronomical units 
(1 a.u. = distance from sun 1o earth). 

2. .Jupiter lias a radius of 74,000 km. A satellite completes an orbit, 
about Jupiter every J 6.7 days. The radius of the orbil of the satellite is 
27 times the radius of the planet. Compute the mass of Jupiter. 

3. What docs Kepler's second law say about the duration of winter in 
the Southern Hemisphere (which occurs in .inly when the earth is farthest 
from the sun) compared with winter in the Northern Hemisphere? 

4. Show the correctness of Kepler's third law of planetary motion 



Motion of Bodies in Space 67 

by equating the centripetal force required to keep a planet in its (circu- 
lar) orbit to the gravitational force due to the sun's attraction. 

5. What is the mass of a planet, .1/,., if it is observed to have a moon 
revolving about it at a distance /?, center to center, in period T? 

6. If the earth, considered to be spherical, were to shrink to 0.9 of 
its present radius, what changes would occur (a) in the length of the 
solar day, (/>) in the value of g at the North Pole, (<•) in the value of g 
at the equator? 

7. The earth satellite Kxplorer III had a highly eccentric orbit witli 
perigee at a height of 109 mi. At this point the velocity was 27,600 ft/sec 
in a direction perpendicular to the radius to the center of the earth. 
Show that this speed is too great for a circular orbit at the radius 
(R, + h) of 4,109 mi. Hence the satellite described an elliptical orbit. 
Its apogee was at the height 1,630 mi. Show that the speed at apogee 
was too small for a circular orbit at radius 5,630 mi. 



It is the supreme art of the teacher to awaken joy in creative expression and 
knowledge. 

A. Einstein, motto for the Astronomy Building, Pasadena Junior College 



Putting on the spectacles of science in expectation of finding the answer 
to everything looked at signifies inner blindness. J, Frank Dobie 



Science is built up with facts, as a house is with stones. But a collection of 
facts is no more a science than a heap of stones is a house. H. Poincare 



Science is organized knowledge. Herbert Spencer 
Science is nothing but perception. Plato 



Travel to Moon 
and Planets 



We first throw a little something into 
the skies, then a little more, then a 
shipload of instruments then our- 
selves. . Fritz Zwicky 



The solar system, consisting of 9 planets moving in elliptical 
paths around the sun, 31 known moons, and many other bodies 
all in motion, does not invite simple straight-line travel from the 
earth to a selected destination. To conserve both power and time, 
departure dates and trajectories must be chosen which utilize 
favorable positions and relative velocities. Conditions favorable 
for return passage may not occur until some time later. 

Owing to the ever-changing distribution of bodies in the solar 
system, no two courses between even the same two bodies are 
likely to be the same. The calculations of desired trajectories 
and corrections of the course while in flight are complex tasks 
for computing machines. 



7.1 INVITATION TO INTERPLANETARY FLIGHT 

Despite obvious difficulties of travel in the solar system, there are 
some interesting favorable factors. The space between the earth 

68 



Travel to Moon and Planets 



69 



and other bodies in the solar system is almost a perfect vacuum. 
This is an ideal environment for a space vehicle to move at speeds 
which make it practical to travel interplanetary distances. Since 
I he earth is one of the smaller planets, it requires a comparatively 
low escape velocity. Its relatively thin atmosphere offers less 
resistance to rapidly ascending and descending objects. The fa el 
that the planets lie in nearly the same plane and move in the same 
sense makes it possible for an interplanetary traveler to apply the 
orbital speed of one planet in launching himself to another. The 
fact that the elliptical orbits of the planets are nearly circular 
means that the energy requirements to transfer a spaceship from 
one orbit to another do not vary greatly for different points of 
departure along the orbit. Finally, most planets rotate in the 
same direction in which they revolve about the sun. So a space- 
ship launched at the surface of one of these planets can get an 
added push by taking oil" in the direction of rotation. 

7.2 LAUNCHING 

Before it is launched, the space vehicle is at the earth's distance 
from the sun, and it is moving with the earth's speed around the 
sun (about 100,000 ft/sec). If launched at greater than the 
earth's escape velocity, the vehicle will take up an independent 
orbit around the sun, at a speed somewhat different from that of 
the earth. 




Fig. 7.1 Launching! to inner and to outer planets. 



70 Looking Out: Rockets, Satellites, Space Travel 

If it is launched in the same direction as the earth's orbital 
motion, the vehicle will have a speed greater than that of the 
earth (l'"ig. 7,1.1), and could reach the outer planets, Mai's, 
Jupiter, etc., if properly directed. The minimum starting speeds 
required to reach these planets arc given in Table 7.1. 

Table 7.1 Minimum launching speeds, with transit times to 
reach the planets 



Plonet Minimum launching 

speed, ftsee 



Transit time 



Mercury 


44,000 


1 10 days 


Venus 


38,000 


150 days 


Mars 


38,000 


260 days 


Jupiter 


46,000 


2.7 years 


Saturn 


49,000 


6 years 


Uranus 


51,000 


1 6 yea rs 


Neptune 


52,000 


31 years 


Pluto 


53,000 


46 years 



From R. W. Buchheim, "Space Handbook," 1958. 

If the vehicle is launched "backward," against the earth's 
velocity, it will move in an orbit like H in Fig. 7.1, so it could reach 
VettUS or Mercury. However, it requires almost, as much energy 
to propel a vehicle in to Mercury as to propel it out to Jupiter. 



7.3 ROCKET GUIDANCE 

In the flight of an unmanned probe, satellite, or missile, one or 
more boosters provide the initial impulse, but after burnout the 
remainder of the flight is unpowered. The vehicle coasts in the 
complex gravitational field of interplanetary space. The accuracy 
of guidance is generally determined by the position and velocity 
at the instant free flight begins. Figure 7.2 gives an idea of the 
maximum allowable errors of angular alignment, and vehicle 
velocity at power cutoff for several kinds of moon-directed 
mis-ions. Inertiul Isiyro) or radio-guidance techniques are ade- 
quate for such relatively simple missions. 

Interplanetary expeditious present complex problems of guid- 
ance. First, a launching site might be chosen at not more than 
2:}° north or south latitude. This is the angle of inclination of the 
ecliptic plane to the earth's equator. The vehicle would be 
launched into a satellite orbit around the earth, in the ecliptic 



Travel to Moon and Planets 71 



S i.o 

§ 
> 

'G 1 

_o 

« 

> 

o a, 

1= I 0.1 

| S. 
.8 o.oi 

E 

§ 



o> 



0.001 



Scientific sotellire (±100 mi) 




Impact on moon (+100 

Around moon 

return to 
braking eclipse 



0.001 



0.01 0.1 

Speed error, per cent 



Fig. 7.2 Maximum permissible errors for alignment of velocity vector and for 
speed at power cutoff. (Genera' Electric, "Space Facts.") 




Initio! ascent -^ ^~v •'/- Earth ' 



^^ 



Fig. 7.3 Possible flight paths to Mars, 



plane, |>'ig. 7.H. With the vehicle in orbit, an ion-propulsion sys- 
tem might be started to cause the vehicle to spiral out into a 
legion where the sun's gravitational field is stronger than the 



72 Looking Out: Rackets, Satellites, Space Travel 

earth's. The vehicle would then be guided into an elliptical trans- 
fer orbit around the sun, planned to intercept the orbit of the, 
destination planet. Where these orbits intersect, the vehicle 
would be directed into an orbit around the destination planet. 

Radio or inertial guidance techniques could serve in the early 
stages of such an interplanetary flight, but would probably 
be inadequate for interplanetary missions of a year or more in 
duration, A three-dimensional form of present-day two-dimen- 
sional celestial navigation may be necessary. A useful instrument 
for establishing a reference direction is the horizon seeker which 
senses the infrared radiation of the earth or some other warm 
body. Optical trackers and magnetometers may also provide 
data to establish the vehicle's position. 

To orient and stabilize a space vehicle, torque is produced, 
either by the ejection of mass (rocket exhaust) or by the rotation 
of a mass within the vehicle. The internal type of torque control 
serves to rotate the vehicle about its center of mass ; it does not 
influence the flight path. 



7.4 RETURN THROUGH THE ATMOSPHERE 

To return safely to earth, a space vehicle must overcome the 
problems of penetrating the earth's atmosphere. There are three 
general types of reentry path, each with its characteristic de- 
celeration pattern: direct descent, orbit decay, and lifting descent. 
These are illustrated in Kig. 7.4. In direct descent into the atmos- 
phere, the maximum deceleration experienced is independent of 
the drag characteristics of the vehicle, but depends on the path 
angle, initial velocity, and characteristics of the atmosphere. 
The altitude at which maximum deceleration occurs does depend 
on the drag characteristics of the vehicle. 

For entry of the atmosphere in orbit decay, the vehicle exe- 
cutes many revolutions about the earth in a very gradual spiral 
that becomes more and more nearly circular. The rate of energy 
loss through aerodynamic drag is sufficiently small so that the 
vehicle's kinetic and potential energies adjust to a momentary 
"equilibrium" orbit, with potential energy decreasing and kinetic 
energy increasing. Thus the velocity of the vehicle actually in- 
creases in the start of orbit decay. The final phase of descent is 
similar to that of direct descent at a shallow angle. 

In a lifting descent, the aerodynamic characteristics of the 



Travel to Moon and Planets 



73 



vehicle are used to obtain a very gradual penetration of the 
atmosphere. The path angle is generally small, a few tenths of a 
degree, and is adjusted to the forces acting on the vehicle. Decel- 
eration increases gradually and can be limited to a relatively 
small value. 

The more gradual the descent, the longer is the time required 
and the longer is the range. Starting at a given altitude and 
velocity, a direct descent may traverse a distance of a few hun- 
dred miles and require about A min. An orbit, decay might cover 
a range of several thousand miles and require 5 to 10 min. A 



Ballistic rocket 



Direct from space 




Fig. 7.4 Different types of atmospheric entry. (Generaf Elec- 
tric, "Space Fads") 

lifting descent from the same point might range over o,000 to 
10,000 mi and require 2 hi*. 

When a vehicle penetrates the atmosphere, the reduction of 
the vehicle's energy is accompanied by an increase in the thermal 
energy of the surrounding air, some of which is communicated to 
the surface of the vehicle. At very high altitude, about one-half 
the energy loss appears as heat in the body. At lower altitudes, 
the heating is produced not directly at the vehicle's surface but 
in the air between the shock wave and the vehicle. Heat is trans- 
ferred from the hot gases of this region to the vehicle by conduc- 
tion, convection, and radiation. 

7 -5 THE NEXT DECADE OF SPACE RESEARCH 

Historically, man's attempts to predict the future of science and 
technology have shown a tendency to be overly optimistic about 



74 Looking Out: Rockets, Satellites, Space Travel 



what will bo accomplished in the immediate future and too con- 
servative about the long-range future. 

It has been predicted that man's curiosity about the unknowns 
of outer space can be only partly satisfied by the placing of 
meters in outer space; eventually he will want to go there to see 
for himself. But it is probable that the extent and pace of space 
research in the foreseeable future will be determined by what 
are regarded as our military requirements. 

The military advantage to be gained from putting man in 
space is at least debatable, From the standpoint of psychological 
warfare, there may be better ways of demonstrating our scientific 
prowess. For man's future happiness, more important pure- 
science experiments might be performed in other fields, such, as 
medicine. Yet many dedicated scientists feel that man-in-space 
experiments are important to our chances of survival 

This viewpoint is stated by Dr. Simon Ramo in the following 
terms. Suppose two rival nations base their security on a race 
for wisdom in the use of limited technical and physical resources. 
Suppose, however, that the first nation makes one decision in 
contrast to that of the second: It decides that man will never be 
needed in space. These two nations start to develop their weapons 
systems of the future. One group has maximum flexibility; the 
other has some prohibitions. "To achieve this maximum of 
flexibility, it is very clear to me that the United States must 
prepare for putting man in space," says Dr. Ramo. 

On the assumption, then, that we shall have military- 
sponsored programs in space technology, one can make some 
predictions for the near future. Many projects involving com- 
munications, weather prediction, manned satellite stations, and 
exploration within the solar system will probably be fulfilled. 

Exploration beyond the solar system now seems unattainable 
chiefly from considerations of time and power. Man's life is short 
when compared with the time required to reach the nearest star, 
even in a vehicle traveling with a speed approaching that of light. 
The other problem, "Where is the energy In he obtained for long 
voyages or to lift large masses into space?" may find an answer in 
the achievement of a nuclear-fusion reactor. It is the thermo- 
nuclear bomb which threatens to make the earth a very unpleas- 
ant place. Ironically, the energy of a controlled thermonuclear 
reaction may provide us with the power resources for a migration 
into space. 



PART 



Looking In: 

Atomic and 

Nuclear Physics 



8 



The Atomic Idea 



Science does not know its debt to imagi- 
nation, R. W. Emerson 



Although Democritus had introduced the word "atom," it was 
the English school teacher John Dal ton (1803) who made fertile 
the assumption that matter is not divisible indefinitely but rather 
is composed of ultimate particles called atoms. Physics dealing 
with phenomena on a scale large enough to be visible to the un- 
aided eye was well understood by the year 1890. Then a remark- 
able mutation occurred in science, caused by the series of dis- 
coveries made in the decade 1895-1905: 

1895 Discovery of x rays by Roentgen 

18% Discovery of radioactivity by lleequerel 

1897 Identification of the electron by Thomson 

H'OO Statement of the basic postulate of quantum theory by 

Planck 
Pi05 Formulation of the theory of relativity by Einstein 

It became clear that the structure of matter was much more 
complicated than had previously been thought. 

The term "modern physics" is often used to designate micro- 
scopic (atomic and nuclear) physics, investigated from the view- 

77 



78 Looking In: Atomic and Nuclear Physics 

point of quantum theory and relativity, as distinct from the 
macroscopic or "classical" physics which was known before 1890. 

8.1 DEVELOPMENT OF THE ATOMIC CONCEPT OF MATTER 

The existence of atoms has been inferred from many experiments, 
the earliest of which were studies of simple chemical professes. 
By 1800, some 30 elements had been identified and the formation 
of chemical compounds had been studied. Lavoisier showed that 
mass appeared to be conserved in chemical reactions. 

Proust, Dalton, Berzelius, and Richter discovered "laws" 
which may be summarized in the statements: 

1. A particular compound always contains the same elements 
chemically united in the same proportions by weight. (Law of 
definite proportions.) 

2. When two elements A and B combine as constituents of more 
than one compound, the weights of B which unite with a 
fixed weight of A (and vice versa) aj-e related to each other as 
the ratios of whole numbers, which are usually small. (Law of 
multiple proportions.) 

Dalton showed that these chemical laws could be explained 
most directly in terms of an atomic theory of matter. Its assump- 
tions arer 

1. All matter is made up of elementary particles (atoms) which 
retain their identity in chemical reactions. 

2. The atoms of any pure substance (element) are alike (on the 
average, at least) in mass and other physical properties, 

:>. Atoms combine, in simple numerical proportions, to form com- 
pounds. 

Dalton's clear formulation of the atomic concept of matter is the 
first important landmark in the development of modern atomic 
physics. 



8.2 AVOGADRO'S NUMBER 



Joseph Gay-Lussac (1808) showed that, at a constant tempera- 
ture and pressure, gases combine in simple ratios by volume. 
Amadeo Avogadro was led (181 1) to make the important assump- 
tion that equal volumes of different gases under the same coudi- 



The Atomic Idea 



79 



tions of temperature and pressure contain the same number of 
molecules. This hypothesis guided Bcrzelius and others in deter- 
mining the ratio of combining weights (e.g., is water HO or H s O?). 

A molecule is the smallest particle of any substance (element 
or compound) as it normally exists. 

An atom is the smallest portion of an element found in a mole- 
cule of any of its compounds. An atom is the smallest portion of 
an element that can enter into chemical combination. 

By measuring combining weights, it is possible to determine 
the relative masses of atoms of various elements. We may arrange 
them in order of increasing mass, assigning a number to each to 
indicate its relative mass. Since only the ratios of the numbers are 
important, we may assign one number arbitrarily to a particular 
atom and adjust the others accordingly. Conventionally, the 
number Hi (exactly) was assigned to an oxygen atom. Then by 
Avogadro's hypothesis, for any gaseous substance 

.Molecular mass of substance 

density of substance 



density of oxygen (Os) 



X 32.000 (8.1) 



Since 19(51, the Commission on Symbols, Unite and Nomen- 
clature in Physics has defined the atomic mass unit as one-twelfth 
of the mass of an atom of the carbon- 1 "2 nuclide. The number 
representing the mass of any atom on this scale is called its 
atomic mass. (The term "atomic weight" is also used.) On this 
scale, which differs only very slightly from the former one, the 
mass of the hydrogen atom is nearly 1 amu and the heaviest 
known atom has a mass of about 250 amu. 

We shall use the symbol .1* to represent, as needed, either 
atomic mass or molecular mass. 

A quantity of any substance whose mass, in grams, is numeri- 
cally equal to its molecular mass is called a mole. In the mks 
system we define the "kilogram mole" as: I kmole of a substance 
is that quantity whose mass in kilograms is numerically equal to 
its atomic (or molecular) mass. The mass of 1 kmole of any sub- 
stance is thus .1 * kg. 

The volume occupied by a mole of any gas is called a gram 
molecular volume. It is 22.4 liters for a gas at 0°C and 76 cm of 
mercury pressure. 

The numerical value of Avogadro's number is not easy to 



80 Looking In: Atomic and Nuclear Physics 



The Atomic Idea 



81 



measure, and it was not known for some time after Avogadro's 
hypothesis was accepted. This constant can be determined inde- 
pendently from experiments in electrolysis, Brownian motion, 
radioactivity, and x-ray diffraction in crystals. The currently 
accepted value of Avogadro's number is 

N A = (6.02486 -f-O.OOOHi) X 10 2S molcculcs/kmolc (8.2) 

Example. Compote the number of atoms in a 1.5-mg sample of lead, 
atomic mass 207. 

The mass of 207 atomic muss units (amu) may be thought of as 
207 kg/mote. Then 

1 5 X I0~ 6 ksr 

N = 207ki7kmole X 6 ° 25 X ^" ^m./kmole 

= 4.36 X I0 1S atoms 



8.3 THE IDEAL GAS LAW 

The gaseous state of matter is simplest to analyze, chiefly because 
the molecules of a gas arc far apart and do not exert appreciable 
forces on each other. The behavior of a gas is expressed by an 
equation of state, which relates pressure, temperature, and 
volume when the gas is in equilibrium. Numerous empirical equa- 
tions have been suggested to describe the behavior of gases. 
The simplest is 



pV - nRT 



(8.3) 



where p = pressure of gas 

V = volume 

T = absolute temperature 

n = number of moles (or kilomoles) of gas present 
In mks units the proportionality constant R is called the gas con- 
stant per kilomole, and from experiment, it has the value 

R = 8.317 X 10 s joules/(kmole){°K) (8.4) 

When other units are used for the variables in Eq. (8.3), the gas 
constant will be expressed differently; for example, 

R = 1.987 cat/ (mole) (°K) = 0.082();> liter-atm/(mole)(°K) 

= 8.317 X 10' ergs/ (mole) (°K) 

Xo actual gas obeys Eq. (8.3) precisely at any nonzero pres- 
sure. Hut this equation holds for all gases when the pressure is 



reduced sufficiently. For then the molecules occupy an insignifi- 
cant fraction of the volume of the container and the widely 
separated molecules exert no attracting forces on each other. 
It is from these considerations that Eq. (8.3) has importance as 
the "ideal-gas" equation of state. 

8.4 KINETIC THEORY OF GASES 

Kinetic theory treats atomic and molecular processes and reaction 
rates by applying elementary methods of mechanics and statis- 
tics. We shall examine what the kinetic theory has to say about 
the observed properties of a gas: its pressure, volume, and 
temperature. 

y\ 







/\ 




/ 


Fig. 8.1 Model for kinetic theory of 
gas pressure. 


West 


i 

i 
i 

i 
i 

i 

i 


— 


East 




A 


/ 



We shall consider a gas confined at a fixed temperature in a 
cubical container with each side of length L. We make the fol- 
lowing assumptions: 



I. 



The molecules have negligible volume; they are "points." 
The molecules move in random directions, but every molecule 
has the same speed v (obviously an oversimplification, which 
we shall reconsider soon). 
The molecules exert forces only in collisions. 
The collisions with the walls are clastic. 

The number of molecules is very large, justifying use of 
statistics. 



The pressure of the gas may be calculated as the force per 
unit area at a wall. Let .V he the total number of molecules in a 
cubical container (Tig. 8.1). Then N/8 will be bouncing lietween 
the east wall and the west wall. Each molecule in this group 



82 Looking In: Atomic and Nuclear Physics 

strikes the east wall v/2L times per second. In each clastic impact 
the velocity of the molecule changes sign and the change in its 
momentum is mv — ( — me) = 2mv, where m is the mass of the 
molecule. From Newton's second law, the force on the cast wall 
is the total momentum change per second at that wall 

„ N i> ,. Nmu % 



The pressure is given by 



P = A = 



J_ Nmjfi _ 1 Nmv* 

l- a£ "a v 



(8.5) 



where V = I* is the volume of the container. 

To compare this prediction with the ideal-gas equation we 
may rewrite Eq. (8.5) as 



pV = i(AW) = \8%m* - %NB t 



(8.6) 



where E k is the translational kinetic energy of one molecule. 
Combining Imjs. (8.3) and (8.0), we have 



nRT = fJVA't 



(87) 



suggesting that the absolute temperature of a gas is proportional 
to the kinetic energy of its molecules. Further, since N/n is the 
number of molecules per kilomole, that is, Avogadro's number 
#,1, we have for the kinetic energy of a molecule 



3 R 

h\ = - — T = : 'I:T 



(8.8) 



The constant k, called Boltzmann's constant, is the gas constant 
per molecule 



k =■ — = 8-317 X 10 a j oule s/(kmole )(°K') 
A', 6.025 X 1 M molecules/ kmole 

= 1.38 X 10" M joule/ (molecule) (°IC) (8.9) 

An improvement can be made in our simples) statement of the 
kinetic theory by removing the second assumption above, that 
all molecules have the same speed. Instead we can say that for 
any particular molecule v- = v t * + vf + v?. If we have a large 



The Atomic Idea 



83 



number of molecules moving at random, the average values of 
iv, tv, and rr are all equal. Equation (8.6) then becomes 

pV = llVmi* 

ami the v of our earlier discussion is replaced by the "rms ve- 
locity," the square root of the mean value of the square of the 
velocity, VP. 



8,5 DISTRIBUTION OF MOLECULAR SPEEDS 

In a gas at a given temperature and pressure, we expect that some 
molecules will have speeds in excess of the root-mean-square 
value, and others will have smaller speeds. Clerk Maxwell applied 
the laws of probability to find the distribution of speeds in a large 
number of molecules in a kinetic- theory gas. He obtained the 
result * 



X 



4 

\Ztt 



(&f« 



e - mT =/s*r f}„ 



(8.10) 



where N = total number of molecules 

N r dv/N - fraction of all molecules whose speeds are between v 
and v + dv 
T = absolute temperature 
fe = Boltzmann constant 
m = mass of a molecule 
For a gas at any given temperature, the number of molecules 
in a speed interval A» increases up to a maximum at the most 
probable speed v p of Fig. 8.2a and then decreases toward zero 
at high speeds. The distribution is not symmetrical about »„, 
for the lowest speed is zero, but the theory predicts no upper 
limit for the speed a molecule can attain. For this reason, the 
average value v of all speeds is somewhat larger than «„. The 
root-mean-square value v rim is still larger. 

As the temperature of a gas is increased, the most probable 
speed of the molecules increases in accord with the meaning of 
temperature (Eq. 8.7). The range of speeds is greater (Fig. 8.2b); 

* L. U. Loeb, "Kinetic Theory of Gases," McGraw-Hill Book Com- 
pany, Inc., New York, 1934; Leigh Page, "Introduction to Theoretical 
Physics," chap. 9, D. Van Nostrand Co., Inc., Princeton, K.J., 1935. 



84 Looking In: Atomic and Nuclear Physics 

~kT 
/itr 



No. 



(a) 



*-/Fs /»=m 




200 



400 



600 



800 



m/sec 




Fig. 8.2 (o) Maxwell distribution of molecular speeds at 0°C. 
(b) Maxwell speed distributions at three different temper- 
atures, T 3 > T-i > T|. 



there is an increase iti (lie number of molecules which have speeds 
greater than a given speed. 



8.6 MEAN FREE PATH 

TIxe mean free patli is defined as the average distance a molecule 
travels between collisions with other molecules. Assume that each 
molecule is a sphere of radius R. Consider the motion of a par- 
ticular molecule among all the other molecules of a gas. It will lilt- 
any molecule whose center lies within a cylinder of radius 2J9 
around its path (Fig. &.',t). In going a distance L, the molecule 
sweeps out a volume ir(2R)' 1 L. If there are >i molecules per unit 
volume in the gas, the moving molecule will bitx(2R) i Ln molc- 



The Atomic Idea 



85 



rules in going a distance L. Its mean free path X is the average 
distance per collision: 



X = 



r(2R)*Ln 4*Rhi 



(8.11) 



This equation is based on the picture of the single moving 
molecule hitting other molecules which are stationary. Actually 
the molecule hits moving targets. The collision frequency is 



2R 



m 




Fig. 8.3 Path of a molecule. 



increased as a result. More complete analysis shows that the 
nieaii free path is reduced to 



X = 



I 



4tt y/2 tr-n 



(8.12) 



Example. I" helium gas at 0°C and I atm pressure, the mean free path 
of one molecule (or atom, He) is 1.86 X 10 -7 m. Estimate the radius of 
8 helium atom. 

The number of molecules per cubic meter under standard condi- 
tions m 



n = 



.V,, 



R = 



22.4 (m 3 . bnole) 
1 



= 2.(59 X 10" molceulcs/m' 



{Aw y/2 Xh) j 



= 1.05 X I0-"m 



which agrees in order of magnitude with other methods of measure- 
ment. Note that the mean free path X is about 1.86 X 10 7 m/2 



86 Looking In: Atomic and Nuclear Physics 

(1.05 X 10 u m) = 900 moleeuliir diameters. From Bq. (S.8) the 
speed of the He atoms is 

u = = — = 1,310 m/sec 
2 m 

So the frequency of collision is 

v 1,310 m/sec 



X 1.86 X 10-' m/(2 X 1.05 X 10 10 m) 
or 7 billion collisions per Becond. 



= 7 X lO'sec - ' 



8.7 MEASUREMENT OF MOLECULAR SPEEDS 

An experimental verification of the. distribution of molecular 
speeds predicted by kinetic theory was reported l>y Stern in 



(a) 



1 



m 



G&rtn 



(4) 



^~=©=" 



Fig. 8.4 Apparatus for measuring molecular speeds, (a) 
Stem's rotating drum, (b) Lammert's slotted disks. 

i926. Atoms (Ilg) from an oven at known temperature pass 
through a slit <S and enter a cylinder C through a narrow slit, in 
its wiill (Fiji. 8.4a). With the cylinder stationary, the molecular 
beam reaches O diametrically opposite the entrance slit. Hut if 
the cylinder is rotated rapidly, the molecular beam is interrupted. 
If a point on (he cylinder wall rotates clockwise from O to () v in 
the time it takes an Tig atom to cross the diameter d, then the 
trace left on the wall by IIg atoms will be displaced counter- 
clockwise a distance 00% (= 0\0) from the reference point. The 
speed of the atoms can be calculated from 

Speed of atoms diameter d 



Speed of drum surface displacement O a O 









The Atomic Idea 87 

In Lammert's method, two disks each with ")0 notches were 
mounted fi cm apart on a rotating axis (Fig. 8.46), in an arrange- 
ment similar to that used by I'izeau to measure the speed of 
light. For a particular constant speed of rotation of the disks, 
only those Hg atoms of a certain speed will be able to pass through 
both notched wheels and reach the collector plate P. !3y varying 
the speed of rotation and by determining the number of atoms 
received at P as a function of their speed, one obtains results in 



N(v) 



u.o 












0,?0 








' - -\ - -1 






0.15 
















0.10 
















^■"1 




0.05 







90 140 



190 240 290 
Speed, m/sec 



340 390 



Fig. 8.5 Speed distribution of mercury vapor molecules at 100°C. (6. 
tammert, Zeitschrift her Physik, 56(3-4): 244-253 (1929).] 



good agreement with Maxwell's predicted distribution (Fig. 
•S..V). (The slight discrepancies with the predicted values, shown 
dotted, were attributed to difficulties of alignment.) 

8.8 SPECIFIC HEATS 

Consider a gas confined at constant volume which is heated. The 
specific heat C? of the gas is defined as the heat required to raise 
the temperature of a unit quantity of the gas one degree. This 
heat is stored in the form of increased kinetic energy of the gas 
molecules. From Bq, (8.8) the increase in the internal energy of 
1 mole divided by the increase in temperature is given by 






(8.13) 



88 Looking In: Atomic and Nuclear Physics 

Thus our basic kinetic theory makes the challenging prediction 
that all gases have the same value of specific heat 

C\ - §[1.987 calV(moIe)(°C)| = 2.98 cal/(mole)(°C) 

The value predicted checks well with experimental data for 
monatomic gases, but not for gases whose molecules are made up 
of two or more atoms. 

Toble 8.1 Specific heats (at 15°C) 



Type of gas 


Gas 


C,. (experimental), 
cal/{moleH°C) 


Monatomic 


He 


2.98 




A 


3.00 


Diatomic 


Hi 


4.80 




o, 


4.96 




N, 


4.94 




CO 


4.95 


Polyatomic 


C0 2 


6.74 




NH 


6.78 




C^Hg 


9.50 



To explain the data of Table 8.1, we may ask whether heating 
may result in energy being stored in forms other than transla- 
tional kinetic energy of molecules, expressible in terms such as 
■frnvf. In a dumbbell-model diatomic molecule (Fig. 8.6b), 
there may be kinetic energy of rotation, expressible in terms 
such as ^/ur. If the two atoms can vibrate and have a force 
constant k, there will be vibrational energy expressible as \l.\r-. 
Each independent mode of absorbing energy is called a degree of 
freedom,/. A theorem of cquipartition of energy, stated by Max- 
well, says that for a large number of particles which obey New- 
tonian mechanics, the available energy is equally divided among 
the degrees of freedom, .]/.V for each. 

Thus modified, our kinetic theory eau be made to agree fairly 
well with experimental data for monatomic gases (/ = 8, C„ = 
35/2) and for diatomic gases (/ = 5, C = 5B/2). One finds 

* The calorie was originally defined as the amount of heat necessary 
to raise the temperature of one gram of water one centigrade degree In 
1948 it was redefined as 1 caloric = 4.1840 joules. The large calorie 
{kcal or Cal) used in nutritional measurements is 1,000 times as large. 






The Atomic Idea 



89 



experimentally (1) that, contrary to kinetic theory, C T varies with 
temperature, and (2) that for polyatomic molecules we need to 
devise empirical models that differ from gas to gas. We have come 
to the limit of validity of classical mechanics when we seek to 
describe the behavior of very small particles of matter (molecules 
and atoms). Quantum theory is the extension of classical theory 
which we need for this {Chaps, I o and Hi). We shall reach another 



Fig. 8.6 Degrees of free- 
dom: independent modes of 
energy absorption. 



(a) O 



(b) 




Degrees of freedom 
3: trans!. 



3 tronsl, 
2 rota . 



3 trans) . 
2 rota . 
1 vibra. 




(c) OmKD 



(d) 



limitation in Newtonian mechanics when we deal with particles 
which are moving very fast (» — * c). Relativity (Chap. 1-1) modi- 
fies Newtonian mechanics in this case. 

PROBLEMS 

1. Copper which has a specific gravity S.9 has an atomic mass 
R3.8 amu. What is the average volume per atom of copper? 

Ans. 1.2 X 10" cm 3 

2. Compute the rms speed at 0°C of the molecules of («) CO», 
(b) H 2 , (c) Xo. Aits. H02 in sec, 1X4 m '.-ee. !()2 m 'sec 

3. If the average distance between collisions of CO* molecules under 
I atm pressure and at Q°C is ft. 29 X 10 6 in, what, is the time between 
collisions? Ans. 1.6 X 10"" sec 



90 Looking In: Atomic and Nuclear Physics 

■i. In a certain electron microscope, electrons travel 1.0 m from 
electron gun to screen. To avoid scattering of electrons by residual 
molecules of nitrogen in I lie vacuum chamber, below what pressure 
would you recommend operating (lie microscope? The radius of a 
nitrogen atom is about 2 X 10 10 m. Ana. p < 8 X 10~ s atm 

5. What pressure will 10 gin of helium exert if contained in a 50-cm* 
cylinder at 2l°C-? Would a cylinder rated at 100 atm maximum safe 
pressure be safe to hold this helium? 4ns. p « 1,000 atm 



Equipped with his five senses, man explores the universe around him and 
calls the adventure Science. E. P. Hubble 



A series of judgments, revised without ceasing, goes to make up the incon- 
testable progress of science. DuClaux 



The main difference of modern scientific research from that of the middle 
ages lies in its collective character, in the fact that every fruitful experiment 
is published, every new discovery of relationship explained . . . Scientific 
research is a triumph over natural instinct, over that mean instinct which 
makes a man keep knowledge to himself and use it slyly to his own advan- 
tage ... To science this is a crime. H. G. Wells 









Wave Motion 



To the mathematician the problems of 
wave motion offer a field for his 
highest power of analysis; to the 
physicist they suggest experiments 
demanding all the skill at his disposal ; 
to the engineer and to those who go 
down to the sea in ships these prob- 
lems are matters of life and death, 
while to the poet and the artist they 
are "the sea dancing to its own 
music " Henry Crew 



In the study of wave motion wc arc concerned with the propaga- 
tion of disturbances in physical systems. A wave is a description 
of a disturbance wliicli propagates from one point in a medium 
to other points, without causing atiy permanent displacement in 
the medium as a whole. Tints sound is a type of wave motion; 
wind is not. 

Wave motion occurs in a medium in which energy can be 
stored in both kinetic and potential form. In an elastic material, 
kinetic energy results from inertia and is stored in the motion of 
91 



92 Looking In: Atomic and Nuclear Physics 

the molecules, whereas potential energy results from the displace- 
ment of molecules against an elastic restoring force. In an electro- 
magnetic wave, we may regard kinetic energy as stored in the 
magnetic field and potential energy in the electric field. In a 
traveling wave, one part of the medium disturbs an adjacent 
part so that kinetic energy at one point is transferred into poten- 
tial energy at an adjacent one, and that potential energy becomes 
kinetic energy at still another point, and so on. 



9.1 TYPES OF WAVES 

A wave is a disturbance that moves through a medium in such a 
manner that at any point the displacement is a function of time, 
while at any instant the displacement at a point is a function of 
the position of the point. The medium as a whole does not pro- 
gress in the direction of motion of the wave. Waves are usually 
described mathematically in terms of their amplitude (maximum 
displacement from equilibrium) and how the displacement varies 
with both space and time. This requires solution of the wave 
equation consistent with the boundary conditions for the particu- 
lar case being studied. In cases most often considered, the wave 
equation is a second-order, linear, partial differential equation. 
The general solutions of the wave equation for a one-dimensional 
space coordinate x are of the form 



* = F(x - vt) + G(x + vt) 



(0.1) 



The functions P and (! are determined by the boundary condi- 
tions, and the speed v by the properties of the medium. The first 
term represents a wave traveling in the positive x direction; the 
second term represents a wave traveling in the negative x direc- 
tion. These are usually sine or cosine waves, for the one-dimen- 
sional case. 

A wave moving on a string is an example of a one-dimensional 
wave. Hippies on water are two-dimensional waves. Acoustic and 
light waves are three-dimensional. 

Waves may be classified in accordance with the motion of 
individual particles. Transverse waves and longitudinal waves 
are the most common types, but there are others. For example, 
as a wave moves on the surface of water, the path followed by an 
individual particle is either a circle or an ellipse. 



Wave Motion 



93 



Elastic waves, of which acoustic or sound waves are a particu- 
lar kind, require a medium having two properties, elasticity and 
inertia. Elasticity of the medium is needed to provide a force to 
restore a displaced particle fo its original position. Inertia is 
needed to enable the displaced particle to transfer momentum 
to a neighboring particle. In an elastic medium one may have, 
in addition to a longitudinal or a transverse wave, a shear wave. 
This is a rotational wave which causes an element of the medium 
to change its shape without a change of volume. 

Light waves, radio waves, and other electromagnetic waves 
are not elastic waves and therefore can travel in free space as 
well as in transparent media. In a vacuum all electromagnetic 
waves travel with constant speed, approximately :j X 10* m/sec. 



-km 



Tvv 



n 



I » . . 




(a) 



(6) 



Fig. 9.1 Wave fronts; (a) plane; (b) spherical. Arrows represent rays. 

In material media the speed is less, and its value depends on the 
medium. 

Waves may be classified further in terms of time: the perio- 
dicity or lack of periodicity of the disturbance. If a long coil 
spring ("Slinky") is stretched out on a table, a single sidewise 
movement at one end will send a pulse or single wave along the 
spring. Kaeh particle remains at rest until the puke reaches it, 
then moves for a short time, and returns to rest, However, a 
continuing to-and-fro motion applied to the end of the spring will 
produce a train of waves. If the motion is periodic, we shall have 
a periodic train of waves. An important special case of periodic 
wave is a simple harmonic wave in which each particle is given an 
acceleration proportional to its displacement and directed toward 
the equilibrium position. 



94 Looking In: Atomic and Nuclear Physics 

An aid in visualizing waves is the idea of a wave front. A wave 
front is a surface drawn through points undergoing a similar 
disturbance at a given instant. The location of a disturbance 
(pulse) at successive e(|iial time intervals may be indicated by 
drawing successive wave fronts. A line perpendicular to a wave 
front, showing the direction of motion of the wave, is called a ray. 
Wave fronts spreading from a point source in a homogeneous 
medium are spherical. Hut at large distance from the source a 
section of the wave front may be treated as practically plane. 

9.2 FUNDAMENTAL RELATIONS 

A wave is commonly identified in terms of either its wavelength 
\ or its frequency J". In any kind of wave motion these two quanti- 
ses are related to the velocity of propagation > by the simple 
equation 

/A = v (9.2) 

The period is the reciprocal of the frequency. The amplitude A 



Wave Motion 



95 



Medium t 

— "-Pi -A* 



Medium 2 



(f 2 >u,) 



(a) -A 



(6) 




Wavelength 



■* — — m- Wavelength \j ■* — 



Fig. 9.2 When a wave passes from one medium to another, in which the wave 
speed is different, (a) the frequency is constant, (fa) the wavelength changes. 






is the maximum value attained by the variable of the wave (e.g., 
the displacement) at a given point in space. 

The frequency of a wave remains constant under all circum- 
stances except for a relative motion between the source of the 
wave and the observer (see Sec. 9.15, Doppler Effect). The speed 
of propagation, however, is dependent on the properties of the 
medium (and, sometimes, also on the frequency). Hence the 
wavelength will vary with speed in accordance with Eq. (9.2), 
as suggested in Fig, 9.2. 

9.3 WAVE FORM 

A wave form is a pictorial representation of a wave obtained by 
plotting the displacement with respect to lime or distance. When 
a wave is traveling along a string in the x direction, the shape of 
the string at some instant I = can be expressed by an equation 



V ■ /(#) when t = 



(9,3) 



which states that the transverse displacement :/ is some function 
f(x) of the distance x along the string. If the wave is moving to the 
right with a speed v, the equation of the wave at some later time 
I is 



y = f{x - Bt) 



(9.4) 



This gives the same wave shape about the point x = vl at time J 
as we observed about point x = at time t = 0. 

The relative positioti (displacement) of two points in a wave is 
called the phase. Two points which have displacements of the 



( = 








t=*t 




a 






b 


\e 


d( 







<vt 



Fig, 9.3 A traveling wave. 



same magnitude and sign (a and h in Fig. 9.3) arc said to have the 
same phase, or to be "in phase." Points c andrf do not have the 
same phase, for although both have zero displacement, the dis- 
placement is decreasing at c, increasing at d. To follow a particu- 



96 Looking In: Atomic and Nuclear Physics 

lar phase in an ongoing wave, wc ask how x changes with t when 
x — vt has some particular constant phase value P. Differentia- 
tion of x — vt = P gives dx/dl = v. So v is the phase velocity 
of the wave. 

A wave form of considerable importance is one defined by a 
sine function 



y = f/o sin -r- (■*•" — i>0 = ij» si" 2» 

A 



(x t) 



(9.5) 



The maximum displacement //« is called the amplitude of the 
wave. The wavelength A represents the distance between two 
points which have the same phase, [■'or a given (, the displacement 
// is the same at x, at x + X, at x + 2X, etc. The period T is the 
time required for the wave to travel a distance of one wavelength 
X, so X = vT, From the second form of LCq. (9,5) it is apparent 
that y has the same value at the times t, t + T, i + 2T, etc., at 
a given position x. 



Wave Motion 



97 










- 








• 


t 


T 
2 




T 





Sine wave 

litt 
y=y sin jr 



Square wave 
«= 1,3,5- 




Saw-tooth wave 

J'0 J'„ r-> 1 tint 

it = 1,2,3- 

Fig. 9.4 Some wove forms, defined in terms of sine functions of the 
frequency, t T, and its multiples, n/T. 



There are many wave forms of interest in physics. To specify 
a particular wave Form, one chooses the appropriate function 
/(.!■)• It is possible to represent any periodic wave form mathe- 
matically as a Fourier scries of sine and cosine terms at har- 
monics (multiples) of the frequency 1, T. Examples are shown in 
Fig. 9.4. 

9.4 WAVES IN A LIQUID 

The waves which we most often see arc those which occur on Ihe 
surface of a body of liquid. Waves also occur within a liquid. 
Their propagation is made use of in marine equipment such as 
the fathometer and Sofar. 







Fig. 9.5 Liquid in a channel, showing two positions of o liquid element being 
considered. 



A quantitative description of a wave motion often can be 
obtained by applying fundamental laws of physics to a particular 
situation. As an example, consider a channel of unit width and 
vertical walls which contains a layer of liquid whose initial depth 
is outlined by the solid line in Fig. 9.5. Examine how this can 
move to successive positions, as suggested by the dotted line. 
We shall apply two physical laws: (1) No liquid disappears or is 
created during the process. 2 The rate of increase <.n momentum 
of any liquid clement must be equal to the net external force 
applied to that element. The force applied to a fluid element at a 
cross section such as .1.1 equals the area times the average pres- 
sure. Since we have assumed unit width for the channel, area = 
I X It. The element considered will be the liquid which initially lies 
under the solid line of Fig. 9.5, between .4.1 and ('('. After time I, 
this same liquid will be under the dotted line between .I'.l' and 



98 Looking In: Atomic and Nuclear Physics 

C'C. The distance from .1.1 to A' A' is u^l, where h, is the speed of 
the liquid lying to tfie left of B. 

To satisfy the first requirement mentioned above 



(At — hi)x = kiUtt 



(9.6) 



where x is the distance the liquid originally at rest lias been accel- 
erated during time interval J. 

To satisfy the second requirement 



f (A, 2 - A.') = "*f -' 



(9.7) 



where ipgki is the average pressure (above atmospheric) in the 
liquid of depth Ai, \pgh-i is the average pressure at CC where the 
depth is As, and p is the mass per unit volume of liquid. The left- 
hand side of Eq. (9.7) is the net horizontal force on the element of 
liquid considered. The mass of liquid ph«x is accelerated from 
rest to speed a 5 in time t, so (ph«x/t)ui is the rate of change of 
momentum. 

The two requirements expressed by Eqs. (9.6) and (9.7) now 
give for the speed m, of the particles of liquid 



, „ (Ai - A;) 2 (Ai + h*)g 



2hih a 



(9.8) 



But the wave speed x/t at which the front of the deeper layer 
advances is given by 



v= i = 4 



(At + A 2 )flig 



2A 3 



(9.9) 



The wave speed v is greater than the speed u of the material 
particles. 

In this simplified treatment, we have disregarded energy )o->i>. 
variation of speed with depth, and a detailed specification of the 
shape of the wave front. Vet we have obtained a valid description 
of the tidal bores which occur in certain rivers. Such a surge 
wave is sometimes employed as a means of dissipating flow 
energy at the bottom of a dam spillway. If the channel is so 
designed that i/ t = ~r then the velocity of the surge relative to 
earth is zero. This form of surge is known us a hydraulic jump. 
it can often lie viewed on a small scale by allowing water to flow- 



Wave Motion 



99 



from a faucet into a basin. The flow can be adjusted so that in 
the basin there can be seen an inner zone consisting of a thin 
layer of water moving rapidly outward. Surrounding this is an 
outer zone which is a thicker, more slowly moving layer. 

The manner in which a continuing oscillatory wave is propa- 
gated may be examined from considerations similar to those just 
suggested for a .-urge wave. In shallow water (say, h ,\ < ,'„ . a 



2(, 



Fig. 9.6 A large-amplitude wove 
steepens to form a bore (Fig. 9.5). 



wave of small amplitude will be propagated without change of 
shape at a speed y/gh, which is consistent with Eq. (9.9), 

If, however, the wave height is an appreciable fraction of the 
liquid depth, the wave speed is significantly greater at positions 
of greater depth. The wave front becomes successively steeper 
(Fig. 9.(i), and a bore starts to form. 

In deep-water waves, individual fluid particles move in 
approximately circular orbits (Fig. 9.7). At the surface, the radius 



StiM-woter leve 





* — "^ Shallow water 



Bottom 



(b) 

Fig. 9.7 Orbital motion of fluid particles for surface waves (a) in deep water, 
and (b) in shallow water. 

of the orbit of a particle is equal to the amplitude of the wave. 
Hut the radius decreases exponentially with depth, and a region 
of almost zero particle motion is soon reached ; hence the behavior 
of the wave is unaffected by the total depth of the liquid. 

lu shallow water there can tie im vortical motion of particles 
at the bottom. The orbits of the particles are ellipses in which the 
vertical axis becomes zero at the bottom (Fig. 9.76). 



100 Looking in: Atomic and Nuclear Physics 

A wave lias equal amounts of potential energy, owing to 
particle displacement above or below the still-water level, and 
kinetic energy, owing to the motion of the particles in their orbits. 
The speed at which energy is transmitted in the direction of wave 
travel is called the group velocity n. us distinct from the phase 
velocity v = x/l. In deep-water waves the group velocity is one- 
half the phase velocity. In shallow-water waves u = e. 



9.5 SOUND WAVES IN A GAS 

In sound waves usually encountered, the intensity is so small 
that the changes in temperature and pressure in the wave are a 
very small fraction of the ambient temperature and pressure. 

Plane wave Front 



«1 
Pi 



«2 
Pi 



Fig. 9.8 Plane wave front in a gas. 



These waves l ravel at a speed which depends only on the ambient 
state of the fluid. 

The propagation of a sound wave in three dimensions can be 
derived from fundamental physical principles starting in this 
way. Imagine, a small prism or a packet of gas enclosed by a 
weightless dcformable membrane. The mass within this packet 
remains constant. The elasticity is expressed by the ideal-gas 
law. The inertia appears in Newton's second law, from which the 
equation for the wave propagation can be derived. 

A simpler procedure may be followed in describing the special 
case of a plane wave front moving from right to left at constant 
speed it, in a gas initially at rest and having density pi. To an 
observer moving with this wave front there will appear to be a 
steady flow of gas from left to rigbl across the wave Ironi (Kg, 
9.8), 






Wave Motion 



101 



Since in a steady flow there can be no accumulation of the 
mass at the wave front, 

Pi*'i = wis (9.10) 

where p« is the density of gas at the right of the wave front and 
h 2 is the velocity of this gas relative to the observer moving with 
the wave front. Also, an increase in gas momentum across the 
wave front requires a drop in pressure from pi to p 2 : 

P2>1? — Pl»l 2 = Pl - P'J (9.11) 

Tiiis expression is obviously related to Bernoulli's theorem for the 
steady flow of on incompressible fluid (p = const). If we consider (he 
fluid flowing past two different cross sections of a pipe at different 
elevations A] and h~ and apply the principle of conservation of energy, 
we get 

(9.12) 



«{As - Ai) + \ («*• - »,*) = p, - pj 

Bernoulli's theorem thus says (hal at any two points along a streamline 
in an ideal fluid in steady flow, the sum of the pressure, I he potential 
energy per unit volume, and the kinetic energy per unit volume have 
the same value. 

For a small disturbance where the fractional changes in gas 
velocity, density, and pressure are much smaller than unity, these 
changes across the wave front can be written as it* = u\ + du, 
Pz = p\ + dp, and p : = pi + dp. When we substitute these in 
Kqs. (9.10) and (t).ll) and neglect product terms of differential 
quantities, we have 

Pl du + tt r dp - (9.10a) 

2p,u, du + urdp = -dp (9.1 In) 

By eliminating du from these two equations, we obtain an expres- 
sion for the wave speed 



u - F p 



(9.13) 



Laplace assumed the compressions and expansions associated 
with sound waves should obey the adiabatic gas law, pp—> = 
constant where y is the ratio of the specific heats, C p /C r . If this 
Relationship for p and p is assumed, the speed of sound becomes 



tti = VyRT = J?l 



(<ut) 



102 Looking In: Atomic and Nuclear Physics 

This result, based on the adiabatic law, does not hold for liquids, 
for gases at extreme pressures and temperatures, or for acoustic 
waves of very high frequencies. However, the pressure fluctua- 
tions in sound waves range from about 10 9 to 10~ 3 atm, which 
justifies the asumption of small disturbance in deriving 
Bq. (9.14). 

9.6 SHOCK WAVES IN A GAS 

In a wave of large amplitude, the wave speed is higher than wi 
in regions of condensation (p > pi) and lower than »i in regions 
of rarefaction. This causes the wave to distort as it propagates. 
Regions of higher condensation overtake those of lower condensa- 
tion (Fig. 9.9). The thin "characteristic lines" are shown for cor- 



Fig. 9.9 When wave speed increases 
with wave amplitude, the wave form 
becomes distorted at successive time 
intervals, 1„, I,, 2f i. 




f D = 



responding points in the wave. The slope di/dx of these lines is 
inversely proportional to the speed. The net effect is to steepen 
compression regions and to flatten expansion regions. Before the 
situation represented at 2d is reached, friction and heat-transfer 
effects counteract the steepening tendency. The compression part 
of the wave propagates without further distortion. It is then a 
shock wave. 

Bomb blasts start as shock waves, large-amplitude compres- 
sion waves. Planes traveling at speeds greater than the speed of 
sound (Much niiniher = speed of body/ local speed of sound > 1) 
generate shock waves which are responsible for the sonic boom 
sometimes heard and felt on the ground. When an astronaut 
reenters the earth's atmosphere, the early motion of his vehicle 
is determined by its shock wave and can be estimated from the 
size and velocty of the vehicle and the known temperature, 
pressure, and density relations for the wave. 



Wave Motion 



103 



9.7 WAVES IN SOLIDS 

Different types of acoustic waves may occur in solids, depending 
on the way in which potential energy is stored in the solid. 

Transverse waves on flexible stretched strings are described 
by an equation of the form 



3P" 6\r a 



(9. If)) 



where ;/ is the displacement of the string at a point ,r. The speed 
of propagation v is equal to the square root of the ratio of the 
tension to the mass per unit length of the string: 



* = 4 In 



(9.10) 



Acoustic waves occur in bars when the bar i> brm and re- 
leased. Here the restoring force is due to the moment of the forces 
about the neutral plane in the bar and depends on the cross- 
sectional dimensions and on Young's modulus. 

Seismic waves which travel through the ground originate from 
natural readjustment of the faults in the earth's crust or from 
explosions. Both body and surface waves result. The body waves, 
which travel through the interior of the earth, may be classified 
into dilationai (longitudinal) waves, which are similar to acoustic 
waves in compressible fluids, and shear (transverse) waves, which 
occur on account of the large shear modulus of most elastic 
solids. 

From known relationships between propagation speeds and 
the mechanical properties of various substances, seismologists 
obtain from seismograms valuable information about the strnc- 
i lire of the earth. Such information can be applied to prospecting 
Tor mines and wells. 



9.8 ELECTROMAGNETIC WAVES 

■lames C. Maxwell recognized about 180-1 that the basic equations 
for electric and magnetic lields could be combined to give an 
equation which resembled the wave equation for mechanical 
waves in a fluid {see Sec. Fi.fi). 



104 Looking In; Atomic and Nuclear Physics 



9.9 SUPERPOSITION OF WAVES 

For many kinds of waves, two or more waves ean pass through 
the same space independently of one another. One can distinguish 
the notes of a particular instrument while listening to a full 
orchestra. The displacement of a particle in the medium at any 
instant is just the sum of the displacements it would be given by 
each wave independently. The principle of superposition states 
that the net displacement of a particle is the vector sum of the 
displacements the individual waves alone would give it. This 
principle holds for an elastic medium whenever the restoring 
force is proportional to the deformation. Superposition holds for 




Fig. 9.10 Analysis of a complex 
wave form. 



electromagnetic waves because of the linear relations between 
electric and magnetic fields, 

The superposition principle does not hold in every ease. It 
fails when the equations describing the wave motions are not 
linear. An acoustic shock wave has a quadratic wave equation; 
superposition does not hold. Hippies which can cross gentle ocean 
swells cannot preserve their identity in breakers. Intermodulation 
distortion occurs in an electronic amplifier when the system fails 
to combine two tones linearly. 

An important consequence of the superposition principle is 
that it provides a means of analyzing a complicated wave motion 
as a combination of simple waves. Joseph Fourier showed that 
any smooth periodic function may be represented as the sum of 
a number of sine and cosine functions having frequencies which 
are multiples of a single basic frequency. The displacement of a 









Wave Motion 105 



particle in the medium transmitting a complex wave is given by 
an equation of the form of 



y = A i sin co( + Ai sin 2ut + At sin Swi + ■ - • 
+ B« + if i cos oil + Bi cos 2U + B,i cos :io>t + 



(9.17) 



In Fig. 9.10, the wave (dotted) which has an approximately 
square wave form is shown to be equivalent to three component 
waves with frequencies in the ratio 1 :3:o and amplitudes in the 
ratio 1 :| : g. The Fourier series representing the square wave is 



A A . 

y = A sin id + tt sin JW + ■=■ sm out + 



9.10 INTENSITY OF A WAVE 



(9.18) 



In any wave, energy is transmitted through the medium in the 
direction in which the wave travels. The amplitude of the wave, 
which is the amplitude of vibration of the particles in the medium, 
is related to the transmission of energy. Each particle has energy 
of vibration which it passes on to the succeeding particles. 

In simple harmonic motion, where there is no damping, the 
energy of a vibrating particle changes from kinetic to potential 
and back, the total energy remaining constant. We may find this 
constant energy from an expression for the maximum kinetic 
energy 



E k - lm(u ui:ix y = 



( 2 r ')' " 



\m@rfv<>y 



2ir % mf 2 y^ 

(9.19) 



where y<> = amplitude of vibration 

T — period 

/ = frequency 

m — mass of the particle 
The energy per unit volume in the medium is the energy per parti- 
cle times the number n of particles per unit volume 



~ - a2* s 8*/W - -iTr-pp-ih? 



(9.20) 



where p = mn is the density. 

The intensity / of a wave is defined as the energy transferred 
per unit time per unit area normal to the direction of motion of 



106 Looking In: Atomic and Nuclear Physics 

the wave. The energy tluil travels through such an area per unit 
time is that contained in a volume which has unit cross section 
and a length equal numerically to the speed 8 of the wave. From 
Eq. (9.20) 



/ = 2Teh>ppij,r 



(9.21) 



The intensity is directly proportional to the square of the ampli- 
tude and to (he square of the frequency of the wave. 

When u wave originates at a point source and travels outward 
through a uniform medium, at some instant the energy is passing 
through the surface of a sphere. A moment later the same energy 
is passing through a larger spherical surface. Since the total 
energy per unit time is the same at the two surfaces, the intensity 
is inversely proportional to the area 4?rr 2 of the surface: 



/ = 



lirr* 



/.": 



(9.22) 



If instead we have a line source (e.g., a fluorescent lamp), the 
energy is spread over successively larger cylindrical surfaces. 
The intensity is inversely proportional to the area 27rr/ of the 
cylindrical surface: 



f = 

2jrrt 



hi 

r 



(9.23) 



Here the intensity is inversely proportional to the distance r. 

For a plane source (e.g., a skylight), which is large compared 
to the distance from the source, the energy passes through suc- 
cessive planes of equal area. There is no divergence of the rays. 
In this case the intensity is independent of distance. 

As a wave passes through any medium, some energy is ab- 
sorbed by the medium. Hence the energy |>;i-<iim through suc- 
cessive surfaces decreases faster than expected from the change 
in area alone. The decrease in intensity due to absorption of 
energy is called damping, A wave whose amplitude decreases for 
this reason is called a damped wave. 

9.11 INTERFERENCE OF WAVES 

The physical effect of superposing two or more wave motions is 
called interference. Where waves arrive in phase, the interfer- 



Wave Motion 



107 



ence is constructive. The amplitude is the sum of the amplitudes 
of the individual waves. Where waves arrive 180° or X/2 out of 
phase, the interference 1 is destructive. 

When two wave trains of different frequency interfere, a series 
of alternate maxim;! and minima is produced in the amplitude of 
the vibration (l-'ig. i).ll). The frequency of these "beats" is the 
difference of the two wave frequencies. A familiar example occurs 
in sound. If two tones of slightly different frequency are sounded 
together, one perceives that the loudness pulsates at the beat 



wwvwm i 1 1 1 1 1 1 1 
wwwvwm i 1 1 1 1 1 1 1 1 

Fig. 9.11 Two waves of different frequency combine to couse beats. Two 
coincidences per unit time are shown for wave trains of frequencies 10 and 12. 

Frequency. Thus if the tones are middle O (2(i!/sec) and O sharp 
(280.5/scc), there will be 16.5 beats sec. 



9.12 DIFFRACTION 

The bending of a wave around an obstacle is called diffraction. 
Diffraction is readily observed as ripples on water bend around a 
stick placed in their path. 

The principles of diffraction and interference are applied in 
the measurement of wavelength of light with an optical diffraction 
grating. A transmission grating is a glass plate upon which is 
ruled many equally spaced lines, usually several thousands per 
centimeter. A parallel beam of monochromatic light falling 
normal to this grating (l-'ig. 9.12) sends waves in all forward 
directions from each slit. Along certain definite directions waves 
from adjacent slits are in phase and reinforce each other. 

Consider parallel rays making an angle with OB, the normal 
to the grating, which are brought to focus at a point P by an 



108 Looking In: Atomic and Nuclear Physics 

achromatic Ions, If ray AP travels a distance X farther than ray 
CP, then waves from .1 and C will interfere constructively at /' 
for they differ in phase by a whole number of wavelength*. The 
wave front CD makes an angle 8 with the grating. From the small- 
est right triangle, the path difference X is seen to be CA sin 0. 
The distance ('A between corresponding points in the ruling is 
called the grating space />. The condition for reinforcement in 
the direction 8 is 

b sin = X (first order) (9.24) 

There are other directions on each side of OH for which waves 
from adjacent slits differ by 2X, 3X, -IX, etc., and for which the 




Fig. 9.12 Diffraction grating. 



corresponding bright images P>, /\, etc., are called the second- 
order, third-order, etc., images. The grating equation in more 
general form is 



b sin 6 = A T X 



(9.2.-)) 



where N is the order of the spectrum and b is the grating space. 
When white light fulls on the grating, it is dispersed into its 
component colors. Spectra are produced at Pi, r\ etc. The dis- 
persion is greater in the higher-order spectra. In each, the colors 
appear in the sequence violet (small X) to red (large X) with in- 
creasing deviation. 

Example. A yellow line and a blue tine of the mercury-arc spectrum h;ivc 
wavelengths of 5,791 A and 4,358 A, respectively. In the spectrum 



Wave Motion 



109 



formed by a grating that has 5.000 lines/in., compute the separation of 
these two lines in the third-order spectrum 

o = ttIsL cm = 5.08 X 10" « cm 
o.OOO 

5,791 A = 5.791 X 10" r ' am 4^68 A = 4.358 X 10 * am 

sinfl = ^ = 3(5^?1_X 10 = <n.O ... „. e _ 2();r 

" b 5.08 X 10-' cm ' 



sin 8i, = 
Separation 



3(4,358 X IP" 6 cm) 

5.08 X 10- 4 cm 
'* — ft ~ 5,3 



= 0.258 ft, = 15.0° 



9.13 STANDING WAVES 

Tf a wave on reaching the boundary of a medium is totally 
reflected., the reflected wave proceeds in the opposite direction 



■fW 




(a) 



(b) 



Fig. 9.13 Standing waves from superposition of waves traveling in opposite 
directions; R is the resultant of A and S. The envelope of a standing wave is 
shown in (M. 

and with equal amplitude (big. 9.13.!. The incident and reflected 
waves add according to the principle of superposition. Two such 
waves, proceeding to the right and left, may be represented by 
the equations 

MM) 

The resultant may be written 

(9.26) 



i/ x = yo sin 



y-i = !/ B sin : 



,'/t + t/s — tf » s i n 



110 Looking In: Atomic and Nuclear Physics 

We may use the trigonometric relation for the sum of the sines 
of two angles 

sin A + sin II = 2 sin l(A + B) cos %{A - B) 

to put Eq. (9.26) in the form 



2wx „ I 
y = 2y sin -y- cos 2w ■=-, 



(9.27) 



This is the equation for a standing (no n progressing) wave. A 
particle at a particular point % executes simple harmonic motion. 
All particles vihrate with the same frequency. But the amplitude 
is not the same for all particles; the amplitude varies with the 
location x. The points x = «X/2 (where n is an integer), at which 
sin (2tx, X) = 0, show no displacement and arc called nodes. The 






amplitude has a maximum value 2i/ n at points .<■ = 



2« + I X 



and 



2 2' 
such points are called autinodos, or loops (Fig. 9.136). 

In general, when a wave reaches a boundary, there is partial 
reflection and partial transmission. Consider a stretched string 
attached to a second string. When a wave in the first string 
reaches the boundary joining the strings, the. reflected wave has 
smaller amplitude than that of the incident wave because the 
transmitted wave in the second siring carries away some of the, 
incident energy. If the second string has a smaller linear density 
than the first, reflection occurs without change of phase. If the 
second string has a greater linear density than the first, there is a 
phase shift of 180° on reflection. From Eq. (9.2fi), it is evident 
that the wave travels more slowly in the denser string. From the 
relation X - v/f, we conclude that in the denser string the wave- 
length is shorter. In a study of light waves we frequently observe 
this phenomenon of change of speed and wavelength as light 
passes from one; medium to another. 

9.14 RESONANCE 

Free or natural oscillation refers to the oscillation of a body or a 
system which has been given a displacement from equilibrium and 
then is not acted on by any external or driving force. The body or 
system will generally have several distinct frequencies of natural 
oscillation. 



Wave Motion til 

If a system which can oscillate is acted upon by periodic 
impulses having a frequency equal or nearly equal to the natural 
frequencies of the system, oscillations will occur with relatively 
large amplitude. This vigorous response of a system to pulses 
nearly synchronous with one of its natural frequencies is called 
resonance. 

Let us determine the natural frequencies of a stretched string. 
When standing waves are established in the string, the end points 
will be nodes. There may be other nodes in between. So the 
wavelength of the standing waves can have many distinct values. 
Since the distance between adjacent nodes is X/2, in a string of 




Vib rotor 



Fig. 9.14 Standing waves in a string driven at a 
frequency nearly equal to a natural frequency, 

length t there must be exactly an integral number n of half 

wavelengths, X 2, so 



X = * 

n 



7i = 1,2,3, 



From Kqs. (9.1) and (9.2ti), the natural frequencies of vibration 
are 



} 21 yitn/t 



n= 1,2,3, 



(9.28) 



These relations may be demonstrated in a string one end of 
which receives energy from a vibrator, such as an electrically 
driven tuning fork. The string passes over a pulley, I' in Fig. 
i>. 14, and is attached to a weight which maintains the string 
under tension I<\ The frequency / of the wave is that of the 
vibrator. The wavelength is 



9 1 i 

/ / V 



F_ 

in I 



112 Looking In: Atomic and Nuclear Physics 

The wavelength may be varied by changing the tension F, which 
changes the wave speed v. Whenever the wavelength becomes 
nearly equal to 2l/n, standing waves of large amplitude may be 
observed. The string is then vibrating in one of its natural modes 

and is in resonance with the vibrator. 

Example, What forte must be exerted on the string, using the apparatus 
of Kg. 9.14, to produce resonance with the string vibrating in one loop? 
The vibration has a frequency 20/sec, the string has a length 18 ft and 
weighs o.O ok. 

From Eq. (9.28), with n = I, 



F = 4 Ifm = 4(18 ft) 



400 6.0 slug 
sec 2 10 X 32 



3-1 11) 



9.15 DOPPLER EFFECT 

There is a change in the observed frequency of sound, light, or 
other waves caused by motion of the source or of the observer, 
\ i ami liar example is the increase in pitch of a train whistle as the 
train approaches and a decrease in pitch as the train passes. 
In the radar system used for traffic control, the speed of a car is 
estimated from the Doppler frequency shift in the radar beam 
reflected from the car. 

In acoustics the Doppler effect deals with cases of relative mo- 
tion between listener and source, plus the effect of any motion of 
the medium. If the source moves toward a stationary observer 
with speed vn, waves emitted with a frequency f s appear to have 
their wavelength shortened in the ratio (u — Vn)/u, because of the 
crowding of the waves in the direction of motion of the source 
(big. 9.15), Theses waves, however, arrive at the listener with the 
speed u characteristic of the medium. 

If, instead, the listener moves with speed u L toward a sta- 
tionary source, the waves appear to him to arrive with speed 
m + v,l- The wavelength in this case is the same as that measured 
when both listener and source are at rest in the medium. 

Xow consider motion of the medium. Let Vm be the component 
of its velocity taken positive in direction from listener to source. 
The velocity components v f , and vs are taken to be positive in 
the direction from listener to source. 



Wave Motion 113 

Then the general equation relating the observed frequency 
fi and the source frequency fs is 



h 



f* 



u + v L — 9u u + vs — I'M 



(9.29) 



There an; important differences between the acoustical and 
the optical Doppler effects. ( I ) The optical frequency change does 




Fig, 9.15 Doppler effect due to motion of the source S toward observer O. 
Wave front 1 was emitted when the source was at position 1; wave front 2 Was 
emitted when the source was at position 2, etc. The drawing shows positions of 
Wave fronts when the source is at S. 

not depend on whether it is the source or the observer that is 
moving with respect to the other. (2) An optical frequency change 
is observed when the source (or observer) moves at right angles to 
the line connecting source and observer. No acoustical frequency 
shift is observed in the corresponding case. (3) Motion of the 
medium through which light waves are propagated docs not 
affect the observed frequency. 

Analysis of the Doppler effect for electromagnetic waves 



114 Looking In: Atomic and Nuclear Physics 

(light) requires use of Lorcntz transformations and the relativity 
postulate that the wave speed c is the same as measured by all 
observers. The result for the observed frequency fa is 



Vl - WJ&) 

j0 Js 1 - (u/c) cos 6„ 

. cos B s + (v/c) 

COS do = . , / ■- } -■; — a 
I + {V/C) COS Or 



(0.30) 

(9.31) 



where 0<, is the angle measured in the observer frame and 9s is 
the angle that would be measured in the source frame if it were 
moving with velocity v relative to the observer frame. 

The term transverse Dtrppler effect refers to the relativistic, 
direction-independent factor in the equations above, fo m 
Is \/l — (u'/c 1 )- This shows that the observed frequency will be 
less than the source frequency regardless of the apparent direction 
of motion of the source. 

The radial Doppler effect is the direction-dependent factor 
and, like the acoustical Doppler effect, is understandable on the 
basis of classical physics, fo « /s/[l — if>/c) cos 0«|. 

In MK-18 II. ]•'„ Ives and (1. It. Stilwcll measured frequencies in 
the spectrum emitted by moving hydrogen atoms and compared 
the frequency shifts with those predicted by the equations above 
for the transverse Doppler effect. This became an experimental 
verification of the special theory of relativity and of the "dilata- 
tion of time" (Sec. 14.!)). 

QUESTIONS AND PROBLEMS 

!. Knergv can lie transferred by particles as well as by waves. How 
can you distinguish experimentally between these methods of energy 
transfer? 

2. When waves interfere, is there a loss of energy? Kxplain. 

3. Why don't wo observe interference effects between the light 
beams omitted from two flashlights, or between the sound waves from 
violins in an orchestra? 

4. A line source (fluorescent lamp) emits a cylindrical expanding 
wave. Assuming the medium absorbs no energy, find how the amplitude 
and intensity of the wave depend on the distance from the source. 

5. A cord 75 cm long has a mass of 0.252 gm. It is stretched by a 
load of 2.0 kg. What is the speed of a transverse wave in this cord? 

Am. 242 m/scc 



Wave Motion 



115 



6. find the speed of a compressional wave in a steel rail whoso 
density is 490 lb/ ft 3 and for which Young's modulus has a value 29 X 10 6 
lb/in.*. Am. 5,200 ft/scc 

7. Compute the speed of sound waves in air at 0°C. The average 
molecular weight of air is 29, y = 1. 40. and R = 8.3 X 10 a joules/ 
(kmole)CK). 

8. If a person inhales hydrogen and then speaks, how will the 
characteristics of his voice be changed? How would the situation be 
changed if carbon dioxide were used? 

9. A student places a small sodium vapor lamp just in front of a 
blackboard. Standing 20.0 ft away, he views the light at right angle-; to 
(be blackboard while holding in front of his eye a transmission grating 
ruled with 14,500 lines, in. He has his assistant mark on the hoard the 
positions of the first -order diffracted images on each side of the lamp. 
The distance between these marks is found to he 14 ft, 2 in. Compute 
the wavelength of the light. 



An ocean traveler has even more vividly the impression that the ocean is 
made of waves than that it is made of water. A. S. Eddington 

False facts are highly injurious to the progress of science, for they often 
endure long; but false views, if supported by some evidence, do little harm, 
for every one takes a salutary pleasure in proving their falseness. 

C. R. Darwin 

What art was to the ancient world, science is to the modern. 

Benjamin Disraeli 



Science and art belong to the whole world, and the barriers of nationality 
vanish before them. Goethe 



-* 



Electric and Magnetic Forces 



117 



10 

Electric 

and Magnetic Forces 



1 cannot help thinking while I dwell upon them that 
this discovery of magnet-electricity (induction) is thB 
greatest experimental result ever obtained by an in- 
vestigator, J. Tyndall 



Electric charges and electric and magnetic forces are important 
in many experiments? designed to reveal the structure and be- 
havior of atoms. All visual information cornea to us in electro- 
magnetic waves, and study of the ultimate structure of atomic 
nuclei depends on electromagnetic processes and detectors. We 
shall outline here only the main ideas in electricity and magne- 
tism needed for our study of atomic and nuclear physics. 

The study of electricity dates from the observation ((500 ux.) 
that bits of straw and other materials arc attracted to rubbed 
amber. The study of magnetism dates back at least as far, to the 
observation that magnetite stones attract iron (but not other 
substances, lieneralh , These two sciences were developed sepa- 
rately until 1820, when Hans Christian Oersted observed a rela- 
tion between them: An electric current in a wire can affect a 
magnetic compass needle. However, the fact that electricity and 

116 






magnetism were initially developed as separate sciences has led 
to some inconveniences in concepts and units which the viewpoint 
of the inks units (which we shall use) seeks to minimize. 



10.1 CHARGE AND MATTER 

We anticipate experimental evidence described in later chapters 
to summarize, some modern basic knowledge. Experiments on 

the electrification or charging of bodies show that there are two 



Nucleons: 
Neutrons O O 

Protons © * 



Nuclei 



Atoms 



:h©(§)i 



Mo I ecu 



Electrons 9© 






'"(©0©) 




Compounds I ( }p ) 

- — ** le 





Visible matter 



Fig. 10.1 Composition of matter. 



kinds of charge. A glass rod may be rubbed with silk, placed in a 
stirrup, and suspended horizontally on a silk thread. If a second 
ulass rod is also rubbed with silk and then brought near the 
rubbed end of the first rod, the two rods will repel each other. 
Hut a hard-rubber rod electrified by rubbing with fur will attract 
the glass ra cJ. Two rubber rods rubbed with fur will repel each 
other. The charges on the glass and hard rubber must be different. 
We add the following details to the atomic picture of Chap. 8. 
An atom has most of its mass concentrated in a very tiny (10 -13 
cm) nucleus. The simplest atom, hydrogen, has a nucleus which 
comprises a single proton. All other nuclei contain, in addition 
to protons, one or more neutrons. Each atom has circulating 



118 Looking In: Atomic and Nuclear Physics 

around its nucleus a number of electrons equal to the number of 
protons within the nucleus. The mass of the electron is about 
1/1,840 the mass of a proton or neutron (Table 10.1). An arbi- 
trary convention adopted in Franklin's time for the sign of the 
two kinds of electric charge leads us to call the electron charge 
negative, the proton charge positive. A neutron lias zero charge. 



Toble 10.1 


Properties 


of some baste particles 




Particle 


Symbol 


Charge 


Mass, kg 


Electron 

Proton 

Neutron 


P 
n 


-e (= -5.60 X 10-"coul] 



9.108 X 10- 31 
1.672 X 10"" 
1.675 X 10 « 



An element may be designated by symbols zEl- 1 , such as «Be* 
for berillium. The atomic: number Z represents the number of pro- 
tons (or electrons) in the atom. Its mass number A represents 
the number of nucleons (neutrons and protons) in the nucleus. 
The number of neutrons is .1 — Z, 

The chemical properties of an atom are determined by its 
atomic number. Two atoms which have the same atomic number, 
but whose nuclei contain different numbers of neutrons, are said 
to be isotopes of the given element. 

Objects can be electrified, or charged, either positively or 
negatively by the removal or addition of electrons. 

Charges of like sign repel ; unlike charges attract. 

In the atomic model proposed by Niels Bohr in 1913, elec- 
trons arc pictured as whirling about the nucleus in circular or 
elliptical orbits. The centripetal force needed To hold an elect nm 
in its orbit is provided by the force of attraction exerted by the 
positive nucleus on the negative electron. 

In addition to the electrostatic (coulomb) forces between 
charges, there are forces which depend on the relative motion of 
the charges. These forces determine the magnetic behavior of 
matter. 

10.2 COULOMB'S LAW 

Coulomb's law (1785) expresses the experimental observation 
that the force of attraction (or repulsion) exerted by one charged 



Electric and Magnetic Forces 119 

object on another is proportional to the product of the charges, 
q x and q 2 , and inversely proportional to the square of the distance 
r between them (where the objects are regarded as "point" 
masses) : 



F = k 



<?><7s 



(10.1) 



The proportionality constant k is a positive number whose value 
depends on the system of units. 

Tn the electrostatic system of units (esu), the unit of charge 
is defined conveniently to make k = 1 in Eq. (10.1): One stat- 
coulomb is that quantity of charge which repels a like charge 
with a force of one dyne when the charges are spaced one centi- 
meter apart in a vacuum (or practically, in air). 

However, the meter-kilogram-second (mks) system of units 
defines a unit for current (ampere) as a fundamental unit; the 
unit for charge (coulomb) becomes a derived unit. The ampere 
is defined in terms of an electromagnetic experiment. The ampere 
is the strength of that constant current which, maintained in two 
parallel, straight, and very long conductors of negligible cross 
section placed in a vacuum at a distance of one meter from each 
other, produces between these conductors a force of 2 X 10 "' 
newtou per meter of their length. The coulomb is that charge 
transferred by an unvarying current of one ampere in one second. 

In principle, we have only to measure the force, in newtous, 
between two 1-coul charges separated by 1 m in vacuum to hud 
k in mks units. The experimental value is 



k = 8.987 X 10° « 9 X 10* newton-m-'/coul 2 



(10.2) 



In the so-called rationalized mks system of units, a different 
constant t , called the permittivity of free space, is introduced in 
the equation for Coulomb's law 



/■' - 



so that 



eo = 



4ttc r 2 



(10.3) 



1 



4irfc 



4tt(8.987 X JO 11 ) newton-mVcoul 2 

= 8.85 X 10 '- coul7newton-m 2 (10.4) 



120 Looking In: Atomic and Nuclear Physics 

The arbitrary inclusion of the Factor 4ir in Coulomb's fundamental 
law makes certain derived formulas more convenient. No it's then 
appear in formulas referring to plane surfaces, a factor 2tt appears 
in "cylindrical" formulas, and 4jt appears in formulas relating to 
spherical symmetry. For example, Table 10.2 gives expressions 
for the capacitance C (charge held per unit potential difference) 
for capacitors of different symmetry as expressed in unratioual- 
ized units and in rationalized mks units. 

The vehemence with which questions of units have long been 
argued may be inferred from Oliver Ileaviside's statement (1893) 



Table 10.2 Comparison of expressions for capacitonce 

Unrationalized 
units 



■ i </ — ^ Plane capacitor 



C = 



Awd 



Rationalized 
units 

C = iAJd 



L 



i i| Coaxial cylinders 
ji (I 



2 In b;a 



C= 



2 In b/a 




Concentric spheres C= — 



Ah 



b -a 



„_ i (4r)ab 
b -a 



that "the unnatural suppression of Aw in the funrationalizedj 
formula for central force, where it has the right to he, drives it 
into the blood, there to multiply itself, and afterward to break 
out all over the body of electromagnetic theory." 

The mks units were adopted by international agreement for 
scientific and engineering use beginning in 1940, but actual 
acceptance of the mks system has progressed slowly. We shall use 
the rationalized mks system of units. 

The statement that Coulomb's law applies to "point" charges 
means, practically, that charges qy and q-> must be associated with 
bodies whose dimensions are negligibly small compared to r. The 
evaluation of the constant k above holds only for the case where 
the two charged particles are in vacuum. If they are immersed in 
some medium, say, oil, the polarization of its molecules greatly 



Electric and Magnetic Forces 121 

diminishes the force between charges r/i and q*. Coulomb's law is 
then written F — <7ifj a /4jrer s , where « is replaced by the larger 
number e, the permittivity of the material in question. 



10.3 ELECTRIC FIELD INTENSITY 

If there are several charges Q u Q->, Q 3 , . . . , in fixed positions, 
and we bring up another charge 7, it will experience a force. We 
say that the fixed charges set up an electrostatic field about them 
and the charge q experiences a force when in this field. We define 
the electric field intensity as the net force per unit + charge 



R 



force 

charge 



+q 



(10.3) 



Electric field intensity is a vector quantity. Its mks units are 
newtons per coulomb (or volts per meter, from Sec. 10.4). We 
can often calculate the value of K at each point of a region of 
space; these values determine the force on (and hence the motion 
of) a charged particle in that region. 



F = gE 



(10.(>) 



The electric field near an isolated point charge Q is easily 
calculated. If a test charge q is brought to a distance r from Q, it 
experiences a coulomb force 



F = q ® 
~ (4x eo )r 2 

The magnitude of the field is then 
P _ Q 



K - 



+q (4areo)r* 



(10.7) 



(10.8) 



This electric field intensity is represented by a vector which, at 
each point in space, points directly away from Q if Q is positive 
or directly toward Q if Q is negative. 

Example. Two charges, q, = —75 X 10 -9 coul and q t = + 75 X 10 -9 

t'tiLil, are 8.0 cm apart in uir. Find I he electric field intensity E at a 



122 Looking In: Atomic and Nuclear Physics 

point P, which is 5.0 era from each charge (Fig. 10.2). The field intensity 
due to 171 U represented by the vector PA and is given by 

2?, _-«L 



4weor i 

= 75 X 10-'(9 .0 X 10') newton 

(0.05)* i-oul 

= 27 X 10* newton/coul 

The field Ei due to charge q* is also 27 X 10* newton/coul in magnitude, 
but its direction is that represented by vector PC. The resultant field E 

9 l =-75xT0" 8 ?,=> +75xicr D 




Fig. 10.2 

is represented by vector PR. Since triangles TKiP and PRC are similar, 
one may write the following proportion: 

PB = DG or _B 8.0 cm 

PC PG 27 X 10* newton/coul 5.0 cm 

E = 43 X 10* newton/coul, parallel to the tine joining q t and q it 

10.4 ELECTRIC POTENTIAL 

In electrical phenomena the concept that is important in cases of 
energy transfer is that of potential difference. If we move a charge 
through an electric field, we exert a force through a distance, and 
so do work. The force exerted at each point is proportional to the 
amount of charge moved, and thus the total work is proportional 
in this charge. The electric potential difference between positions 
b and a is the work done per unit + charge in carrying charge 
from a to b 

AV= V„ - V a = ^ (10.9) 

Potential is measured in volts: 1 volt = 1 joule/caul 



Electric and Magnetic Forces 123 

Considering a field due solely to a fixed charge Q, we shah 
compute the work done by an external agent in bringing another 
charge q from a great distance (infinity) in to P at a distance 
H from Q. As q is moved an infinitesimal distance ds along an 
arbitrary path (Fig. 10. .'5), it will be acted on by a practically 




Fig. 10.3 Calculation of potential at a point P. 



constant force F = qQ/4irt D r s . The work done on the system of 
charges by the agent exerting force F is 



dW = F cos 6 ds = -7-^, dr 



(10.10) 



where the negative sign comes from the fact that F and ds 
cos fl (= dr) are vector quantities in opposite directions. The 
total work done in bringing q from =o to R is then 



IK = ~ ( lQ f R< t= flQ 
4ir«o J" r s hrtjt 



(10.11) 



Dividing by q in Eq. (10.1 1), we have the work per unit charge, 
which is the potential, given by 



v R - r„ = 



Q 



4aW£ 



(10.12} 



Strictly, we have defined only difference of potential. Jf arbi- 
trarily the potential is taken to be zero at infinite separation of 
the charges (l'» = 0), then the potential (sometimes called the 



124 Looking En: Atomic and Nuclear Physics 

"absolute" potential^ at a distance r from an isolated point charge 
Qis 

Q 



V - 



4xe r 



(10.13) 



Tf q and Q are both positive (or both negative) charges, then 
the external agent bringing </ toward Q exerts a force in the direc- 
tion of the motion and does a positive amount, of work on the 
system. II" k(ijQ '"). This work is stored in the system as poten- 
tial energy 

qQ 



P = 



4vt r 



= qV 



(10.14) 



Tin's potential energy can be recovered. If the charge q at a point 
distant r from Q is released, it will fly off; its potential energy is 
converted into kinetic energy. 

If q and Q have different signs, then in the trip from «; to It, 
the agent will have to hold q back (to prevent acceleration). 
The IF of Eq. (10.1 1) will be negative. Because energy is trans- 
ferred from the electric field to the agent, the charges are placed 
in a configuration of lower potential. Energy would have to be 
put back into the system to separate the charges again to infinity. 
These ideas will be used in calculating the energy stored in an 
atom of hydrogen, where a positive nucleus attracts the negative 
electron (('hap. lo). 

lisniiifih-. Kleetrons which leave a healed filament with negligible energy 
arc accelerated to pass through an aperture in a metal plate maintained 
at. a potential of !)00 volts above thai of the filament. What is the final 
s))eod. of the electrons? 

Each electron has a charge of — 1 .130 X 10"" coul and a mass of 
9.00 X 10" 11 kg. The electron gains kinetic energy equal to the work 
dime on it in falling through potential difference V. 

(l)»w* = Ve 
Hence 



">/?-( 



2 X 1.6 X lO" 1 ' coul 901) 



9.11 X 10-" kg 



*)'= M X 



I0*m/sec 



The kinetic energy attained by an electron in falling through a 
potential difference of 1 volt is given the name electron volt (ev). 

1 ev - (-e)(-AV) = 1.602 X 10- ,u coul (1.00 joule/conl) 
= 1.002 X 10- '• joule 



Electric and Magnetic Forces 125 

10.5 ELECTROSTATIC DEFLECTION 

The deflection of charged particles by electric and magnetic fields 
has been important in the identification of elementary particles 
and in the development of such useful devices as the cathode-ray 
oscilloscope and the mass spectrograph. 

As a special case, consider a parallel-plate? capacitor (Fig. 
10.4) with charge —Q on the upper plate and +Q on the lower 



Fig. 10.4 £ = AV, s for o uniform 
electric field. 



z 



I 



+ Q 



plate. If the distance s between the plates is small compared with 
the other dimensions, the electric field E is uniform in the region 
between the plates. If we lake a small charge </ from the upper 
plate across to the lower, the work done is the product of the con- 
stant force liq and the distance s. From the definition of potential 
difference, the work is also the product" of the charge q moved and 
AV. By equating these, A'r/s = q AV, we have 



E - 



AV 



for uniform field 



(10.15) 



A device for studying the charge and mass of particles con- 
sists of an evacuated tube in which a narrow beam of particles, 
defined by slits c, and <•■. passes between the plates of a parallel- 
plate capacitor and then impinges on a fluorescent screen S' where 
it produces a visible spot (Fig. 10.;">), The x component, of the 
velocity of a particle suffers no change as the particle passes 
through the capacitor and goes to the screen. In the electric field 



P 



1 







—TT 






AV 












+ 








k-L 


-4— 







S' 



Fig. 10.5 Electrostatic deflection of a beam of charged particles. 



126 Looking In: Atomic and Nuclear Physics 

of the capacitor, a positive particle will experience an upward 
acceleration n tf from tlie force exerted by the field E on the charge 



F v _ qE 
m m 



(I O.K.) 



The particle emerges at the right side of the capacitor with 
velocity components 



v x = v 



I 



m m v 



(10.17) 



where ( = l/v, the time required for the particle to pass through 
the capacitor. 

The particle emerges from the capacitor at an angle 8 with its 
original path, where 



. ? T.' I 



ten b . "JL = ■' E 
v x m 



(10.18) 



The deflection A C observed on the screen is the sum of the deflec- 
tion AB which the particle incurs while in the capacitor and 
deflection BC brought about by the v„ velocity component while 
the particle travels distance D. 



2 m ir 
BC = Dt&n0 = ^E-,D 



(10.19) 
(10.20) 



Thus the measured deflection AC is as if the beam were abruptly 
deflected through an angle midway through the capacitor: 



AC = AB + BC = \ q —,+ (>E { - (d + -A tan 8 (10.21) 
2 mv* mo- \ 2/ v ' 

If we measure AC, D, and I, we can find tan 6. Measurement 
of AV and $ determines E and, through Eq. (10.18), gives a value 
for q/mv 1 . If we know the initial speed v of the particles, we can 
iind a value for q/m, or vice versa. The experiment does not 
determine q and m separately. The experiment is usually done in 
such a way that is a very small angle, so that tan = (ex- 
pressed in radians). Then, for a given instrument, is inversely 






Electric and Magnetic Forces 127 

proportional to the kinetic energy of the particle. Such deflection 
experiments are important in identifying, sorting, and utilizing 
charged particles. 

Toble 10.3 Charge-moss ratios for 
several particles 



Particle 



q/m, coul kg 



Electron 
Proton 
« particle 



•1.75V X I0 11 
9.579 X 10 7 
4.822 X 10 7 



10.6 ELECTRIC CURRENT 

Electric charges in motion constitute an electric current. In 
metallic conductors there are many "free" electrons, that is, 
electrons not bound strongly to particular atoms of the metal. 
Each electron moves in an irregular path, continually colliding 
with atoms of the metal. If a wire is connected across a battery, 
an electric field is set up within the metal. The electrons tend to 
drift from regions of low potential to regions of high potential. 
This electron "wind" is the current. The continual collisions are 
responsible for the resistance of the metal. The kinetic energy 
gained by the electrons from the field and given up in collisions 
is l he power loss i-R which produces heating in any current- 
carrying conductor. 

In electrolytic solutions, in some types of vacuum tubes, and 
in certain solid-state devices, electric current may result from the 



Positive plate +. — Negative plate 
|t 



,/) Ammeter 



Electron flow 



High potential 



Conventlal current 



Low potential 



Fig. 10.6 Direction of electron flow and of con- 
ventional current. 



128 Looking In: Atomic and Nuclear Physics 

motion of both positive and negative charges. Any currenl direc- 
tion is a convention; the choice of sign is arbitrary. In this honk 
we shall regard the direction of conventional current as that of 
the flow of positive electricity. The conventional current is from 
high potential to low potential ("from + to — "} in the external 
circuit. 

If two parallel wires carrying current in the same direction are 
brought near each other, they attract each other. This effect is 
the basis for the definition of the ampere, the inks unit of current 
(Sec. 10.2). This attraction is not an electrostatic (coulomb) force 
between unbalanced charges. It is a magnetic force arising from 
the motion of charges. It is convenient to discuss these magnetic 
forces in terms of a field. 

10.7 MAGNETIC INDUCTION 

The basic vector for describing a magnetic field is called the mag- 
netic induction, B. (Magnetic field strength would be an appro- 
priate name for B, but historically this name has been assigned 
to another vector H connected with magnetic fields.) We identi- 
fied and measured an electrostatic field in terms of the force 
exerted on a unit positive charge. Kxperimetitally we identify the 
presence of a magnetic field from the fact that if a magnetic field 
is present, a moving electric charge will experience a sideways 
magnetic force. 

The magnetic induction H is defined as the vector which satis- 
fies the relation 



F = q(v X It) 



(10.22) 



where force F, charge q, and velocity v are the measured quanti- 
ties. This notation of a "vector cross product" means that F is 



Fig. t0.7 F = q(v X B) 




Electric and Magnetic Forces 129 

perpendicular to both v and B and directed so that, if the fingers 
of the right hand are directed from the direction of v (around 
through an angle of less than 180°) to that of B, the right thumb 
will point in the direction of F. The magnitude of F is given by 

F^qvIixmB (10.23) 

where 8 is the angle included between the positive directions of 
v and B, Xote that a vector cross product v x B is zero if v is 
parallel or antiparallcl to B, Notice also that A x B = — B x A; 
that is, A X B is equal in magnitude but opposite in direction 
to B X A. 

The unit for B, from Eq. (10.22) is newton/(coulomb meter/ 
second). This inks unit for B is given the name weber/meter 1 



we her _ newton 
m 2 eoul m/sec 



- 1 



newton 



amp-m 

An earlier cgs unit for H, still often used, is the gauss. 
weber 



1 



nv 



= 10* gauss 



The summation of B over a surface J'B ■ r/s is called the magnetic 
flux *. The weber is the unit of flux. 

When a charged particle moves through a region in which 
both electric and magnetic fields are present, the resultant force 
on the particle is given by 

F = ryE + f/(v X B) (10.24) 

Only in the special case where E, v, and B are suitably oriented 
can we replace Eq. (10.24) by a scalar equation which suggests 
straight, addition: 

F = q E + qv B sin B (10.25) 

where is the angle included between the positive directions of 
v and B. 



10.8 MAGNETIC FORCE ON A CURRENT 

An electric current may be visualized as moving charges. Assume 
that in a conductor of length I there are n conduction electrons 
per unit volume, each with charge q and each having an average 
drift speed i\ (The negative electrons drift in a direction opposite 



130 Looking In: Atomic and Nuclear Physics 

to that of the current, Fig. 10.8.) The distance an electron moves 
per second is v. The volume of charge passing a certain cross sec- 
tion A of the wire is Av. In this volume there are nAv = JV con- 
duction electrons. If this conductor is in a uniform magnetic 



/ 






U 



5 



Fig. 10.8 Representation of an 
electric current. 



induction B, the force q(v xB) on each moving charge in the 
conductor produces a force on the conductor of length / which is 

F = Nq(v X B) 

But the velocity is 1//, and Nq/t is the current i, so the equation 
for the force hecomes 

F = Nq (j X b) = ( N f\ (1 X B) = »(| X B) (10.26) 



or 



F = Bil sin d 



10.9 MAGNETIC DEFLECTION OF CHARGED PARTICLES 

The force on a charged particle moving in a magnetic induction 
is at right angles to B and to v. The particle is accelerated, I nit 
always perpendicular to its velocity. The magnetic force changes 
the velocity (vector) but not the speed (scalar). No work is done 
on the particle by the magnetic force, for cos 9 = in the expres- 
sion W = /F cos da. 

When a charged particle enters a uniform held with its ve- 
locity perpendicular to B, the particle experiences an acceleration 
of constant magnitude qvB/m perpendicular to its velocity. The 
particle describes a circular path with constant speed. The cen- 



Fig. 10.9 Path of charged particle in plane 
normal to 1i. 




Electric and Magnetic Forces 131 

tripetal force needed to keep the particle in circular motion is 
supplied by the magnetic side thrust. Since the velocity of the 
particle is always perpendicular to the induction, sin 8 = 1 in 
Eq. (10.25). Newton's law F = ma then can be expressed as 

The momentum of the particle mv can be found if we know B 
and q and measure r. If measurements of electrostatic deflection 
and magnetic deflection are carried out on the same beam of 
charged particles, one can determine both q/m and v for the 
particles. In this way Thomson measured the charge/ mass ratio 
for electrons in 1897 (Chap. 11). Similar deflection methods arc 
used today in some types of mass spectrometers to obtain accu- 
rate values of q/m for ions and isotopes. 



10.10 MAGNETIC INDUCTION OF A CURRENT 

We have just considered problems relating to the forces exerted 
by a magnetic induction on a moving charge or on a current- 
carrying conductor. A second class of problems involving mag- 
netic lields concerns the production of a magnetic induction by a 
current-carrying conductor or by moving charges. 

The relationship between current i and magnetic induction B 
is given by Ampere's law. The magnetic induction at a point P 
arising from a current i in a wire is the vector sum of contribu- 
tions from every element of the wire. The induction at P due to 
the current in element d\ of the wire is 



d\\ - I \ -_ til X r 

Air r' 



(10.27) 



We have again used the notation of the vector cross product. The 
magnitude of rfB is given by 



dB - 



idl 
4w r 2 



u., 



sin 8 



(10.28) 



The constant u,. is called the permeability of free space. In the 
mks system its value is 

/in = 4jr X 10 -7 joule/amp*-m 



132 Looking In : Atomic and Nuclear Physics 

The direction of r/B is perpendicular to the plane of the vectors 

dl and r, and such that if the fingers of the right hand arc turned 
(through an angle less than 180°) from dl to r, then the right 
thumb will point in the direction of dli. 




Fig. 10.10 Induction dU contributed by cur- 
rent element i dl. 



Ampere's law as expressed in the last two equations cannot 
be subject to direct experimental check, for we cannot isolate an 
element i dl of an electric circuit. Actually Ampere's law was not 
deduced from any single experiment. Rather it summarizes many 
experiments dealing with magnetic effects of circuits of different 
geometry and witli magnetic forces exerted by currents on each 
other. 

To illustrate the tise of Ampere's law we shall calculate the 
magnetic induction (I) at the center of a circular loop, and (2) 
at a point near a long straight conductor. At the center of the 
loop (Fig. 10.11) the direction of the magnetic induction H is 



Fig. 10.11 Induction B (out of page) at 
center of a circular loop. 




Electric and Magnetic Forces 133 

perpendicular to the plane of the current elements i dl and r, in 
the sense given by the cross product d\ x r, or out of the page. 
Since a radius to any point on the loop is perpendicular to the 
tangent to the circle at that point, sin = 1 in Bq. (10.28). 
Writing r d<p for dl, the magnitude of B at the center of the loop is 

B = toL-* r*"= ■>,■ 0&»> 

To calculate the magnetic induction at a distance R from an 
infinitely long straight wire (Fig. 10.12), it is convenient to take 



Fig. 10.12 Induction B near a Mroight conductor. 




the z axis along (he wire in the direction of / and with the origin 
O at the point of the wire closest to l\ The upper and lower halves 
of the wire make equal contributions so we may compute B by 
integration of the expression 



B = 2 f- u i 



/-*. 



sin 



'/: 



4x } - » r 
For the section of wire below the origin (z < 0) we have 



(10.30) 



-R 



— tan 8 



and 



R 



— sin 6 



dz = d(.^\=«4 

\tm 0} sin 2 



de 




I 



Equation (10.30) may be written in terms of one variable 0: 
R _ tint [*ft . . sin 2 e R d0 noi r*te . 



134 Looking In: Atomic and Nuclear Physics 
giving 



B = 



li.nl 

2-xii 



(10.32'] 



10,11 DIRECTION RULE 



The relative directions of the vector quantities in Eqs. (10.22) 
and (10.27) implicit in the vector cross products can be remem- 
bered conveniently from the following rules: 

(a) If, in imagination, the right hand grasps a current- 



=^z^ 




(a) 



(M 



Fig. 10.13 Magnetic induction B near an electric 
current: {a} Side view, i toward right; (b) end view, 
j out. It counterclockwise. 



Current "in" 



Current "out" 



B 



B, 



© 



(a) 





(b) 



(c) 



Fig. 10,14 Force on a current-carrying conductor is from strong field region 
toward weak field region. 



carrying conductor with the thumb pointing in the direction of 
the conventional (+) current, the fingers encircle the wire in the 
same sense as the magnetic induction B. Thus in rig. 10.13a, B 
is out of the page above the wire (indicated bj dots) ; B is into 
the page below the wire (indicated by crosses). 



Electric and Magnetic Forces 135 

(h) When a current-carrying wire is in a magnetic field, the 
magnetic force on the wire is directed from the region of stronger 
induction toward the region of weaker induction. 

Consider a wire perpendicular to the page carrying a current 
into the page (Fig. 10.14a). The local magnetic induction encir- 
cles the current in the clockwise sense. If now this current-carry- 
ing conductor is placed in an external field B, (Fig. 10. 1 4b), 
B and B\ reinforce each other above the wire (strong field region) 
and partly cancel each other below the wire (weak field region). 
The force F on the wire is down. A representation of the net field 
due to B and Bj is shown in Fig. 10.14c, 

10.12 INDUCED CURRENTS 

A further important relation between magnetic fields and electric 
current is the principle of induced emfs on which the design of 
generators, transformers, and motors is based. Michael Faraday 
and Joseph Henry, at about the same time (1831), showed that 
an emf is induced in a conductor when there is any change of mag- 
netic flux linked by the conductor. It is convenient to consider 
this single principle from two viewpoints, (i) An emf e is induced 
whenever a conductor moves across a magnetic field 



e = IvB sin d 



(10.33) 



where I = length of wire 

v = its velocity 

— angle between » and B 
When mks units are used on the right of Eq. (10.33), e is given 
in joules per coulomb or in volts. 

(2) An emf is induced whenever the flux ($ = BA) changes 
through a circuit: 



e = — 



d* 
dt 



(10.34) 



I f the rate of change of flux is in webers per second, the emf e is 
in volts. 

Lena's law states that whenever an emf is induced, the induced 
current is in such a direction as to oppose (by its magnetic effects) 



136 Looking In: Atomic and Nuclear Physics 

the change inducing the current. Lena's law is really an example 
of the conservation of energy principle. 

As applied to Fig. 10. l. r >, Lea* 's law says that the direction of 
current induced in the moving wire must he Mich as to oppose its 
motion. This requires the magnetic force on the wire to be toward 



Fig. t0,!5 Emf induced in wire ob 
moving across a uniform magnetic 
field. 



the left. From Sec. 10.1 1 b, the net induction ahead of the wire (at 
right) must be greater than that behind the wire (at left). To 
reinforce the external li, directed out, ahead of the wire, the 
induced current must be down, from a to b, from tint right-hand 
rule (Sec. 10.1 In). 





a 


II 




b 




Fig. 10,16 Emf induced in a coil moving 
from a to b in nonuniform magnetic field. 



Note that in this "generator" b is at a higher potential than a. 
Positive charge is forced to flow from a (low potential) to b (high 
potential) by the work an external agent does in moving the wire 
ab against the magnetic force. In the external circuit bca, the 
charge flows from high potential toward low potential ("from 
+ to — "); it can do useful work, and it produces ,-/>' heating in 
the conductor. 



Electric and Magnetic Forces 137 



QUESTIONS AND PROBLEMS 

1. State some similarities and some differences between the phe- 
nomena of electric fields and gravitational fields. 

2. A positively charged rod is brought near a ball suspended by a 
silk thread. The hall is attracted by the rod. Does this indicate that the 
ball has a negative charge? Justify your answer. Would an observed 
force of repulsion be a more conclusive proof of the nature of the charge 
on the ball? Why? 

3. The circuit in the diagram consists of two concentric circular sec- 
tions AC and l)E and two radial sections CI) and HA. There is a cur- 



Fig. 10.17 




rent i in the direction shown. Starting from Kq, (I0.2S). derive an 
expression for the flux density at O, the common center of the arcs. 
Show the limits of integration arid any special values of factors in die 
equation. Show clearly the contribution of each part of the circuit. 

What is the direction of the (lux density at 0? Ans. ? ol(fii ~ _ ^ij 

4. A 5.0-m straight wire ab (Fig. 10.18} is allowed to fall through a 
uniform magnetic induction of 2.0 XII) 5 webcr/m* directed pcrpen- 



Fig. 10.18 



x 
a 



V 



dicular to the wire, (n) What is the emf induced in the wire at the 
instant its speed is 3.6 m/see? (6) What is lire direction of this emf in 
the wire? (c) Which end of the wire is at the higher potential? 

Am. 3.6 X I0-* volts; toward right: 1',. > l'„ 

5. A Ferris wheel is 100 ft in diameter. Ms axis of rotation i- on a 
north-south line, (a) If the horizontal component of the earth's field is 
2.00 X 10 _i weber/m* and the wheel is rotating at 2.00 rev/rain, what 
i* the potential difference existing between the axle and the end of one 
apoke? (b) Which i^ at the higher potential? Ans. 488 pv 

(}. Assume thai this mom has a uniform magnetic field directed 
vertically downward and of flux density B = 1.0 X 10-" weber/m*. 
(a) Determine the magnitude and direction of the force on an electron 



138 Looking In: Atomic and Nuclear Physics 

thai enters the room moving due east and at an angle of 30° above the 
horizontal with a speed of 3.0 X 10* m/sec. What, will be the speed of 
this electron 1.0 X 10 '■' see after if enters the room? (b) An east-west 
wire is stretched horizontally across the room. What will be the direction 
and magnitude of the force on :i (i.O-m section oT the wire when there is 
a current, of 12 amp westward in the wire? 

Am. 2.4 X 10-" newton, south; 3.0 X 10* m/sec; 7.2 X 10" 1 newton, 
sou til 






The most important thing a young man can learn from his first course in 
physics is an appreciation of the need for precise ideas. W. S. Franklin 



A new principle is an inexhaustible source of new views. 

Marquis de Vauvenargues 



To succeed in science it is necessary to receive the tradition of those who 
have gone before us. In science, more perhaps than in any other study, the 
dead and the living are one. Charles Singer 



11 



The Electron 



The electron has conquered Physics, 
and many worship the new idol rather 
blindly, H. Poincare 



It is the purpose of this chapter to discuss the experimental evi- 
dence of the existence of the electron, some of the measurements 
which have been made on it, and the limitations of the classical 
free-electron theory of the conduction of electricity in metals. 

11.1 IDENTIFICATION OF THE ELECTRON 

In a paper "On Cathode Rays" (1897), J. J. Thomson first estab- 
lished the existence of free electrons. Thomson's investigation was 
prompted by ihe divergent opinions people had at that time about 
the nature of cathode rays. Experimenters had shown that at 
low pressures (about 0.01 mm of mercury) air becomes a good 
conductor of electricity and that, the discharge of electricity 
through a gas produces light whose color depends on the gas and 
in a pattern which depends on voltage and gas pressure (Kig. 11.1). 
Some people considered the rays charged particles; others 
thought of the display as a phenomenon in the "aether." Thomson 
139 



140 Looking In: Atomic and Nuclear Physics 

suggested that an explanation based on particles as :i working 
hypothesis was more likely to he successful and could he more 
easily tested hy known laws (of mechanics) than any explanation 
based on properties of the aether about which little was known. 
Mr therefore devised experiments "to test some of the conse- 



Crooke's dork space 
Cathode / _ 




Faraday dork spoce 

7 



#- !■ % 



Positive column 



-A 



Anode 



Negative glow II Anode glow 

To pump 



Cathode glow ' 
Fig, 11.1 Discharge of electricity through a gas at reduced pressure. 



quences of the electrified particle theory." The objects of Thom- 
son's experiments were: 

1. To verify that cathode rays carry a charge, a charge which 
accompanies the rays when they are deflected by a magnetic 
field 

2. To investigate quantitatively the deflection of cathode rays 
in an electric field, which deflection also indicates the presence 
oT charge 

:?. To determine the energy of the cathode rays and, by using 
this value with data on the magnetic deflection, to deter- 
mine the speed and the ratio of charge to mass e/tn, for the 
"particles" 

4. To determine speed and <■ hi also from a combination of elec- 
tric and magnetic deflections 

Thomson obtained information on each of these properties in 
the following ways: 

1. In a tube such as shown in Fig. 11.2, cathode rays leave 
the cathode C, pass through an opening in the anode A, and reach 
a region B where they are deflected by a magnetic field, pass 
through an opening in a grounded cylinder 6, and finally reach a 
collecting conductor /' mounted inside that cylinder. An increase 
in charge is registered by an electrometer connected to /' only 
when rays enter the opening in Q. The observations prove that a 
charge is inseparably connected with the cathode rays and that 
it is a negative charge. 



The Electron 



Mi 



2. Deflection by an electric field was investigated in the tube 
shown in Fig. 11.3, the precursor of our modern cathode-ray 

oscilloscope. It verified the negative charge of cathode rays. 

3. The energy of the cathode rays was measured in a tube 
without deflection plates and provided with a screened electrode, 



Fig. 11,2 Discharge tube far demon- 
strating the (negative! charge and 
magnetic deflection of cathode rays. 
The magnetic field is excited in space 
S by coils placed outside the tube. 




as in Fig. 11.2. The innermost, electrode contained a thermo- 
couple. Its increase in temperature in a given time was measured, 
and simultaneous measurement was made of the charge from the 
rays received at the thermocouple. 

If, in the time considered, A r particles strike the thermocouple 
each bearing a charge r, the total charge is Q = A>. From the 
ri.-e in temperature the total energy is known 

Ek - S\mt:~ (i i.|) 




Fig. 11.3 Thomson's tube for electricol deflection of cathode rays. 



142 Looking In: Atomic and Nuclear Physics 

where m is the mass and v the speed of one particle. Next the 
radius r is measured for the path described by the particle in a 
magnetic field. The centripetal force is equated to the magnetic 
thrust 



™ = cvB 

T 



(11.2) 



For each of the two flat circular coils used to produce the mag- 
netic field, B is given by Eq. (10.29) 



o _ Mo/ 
B - "27 



(11.8) 



where J is the current and mu is the permeability of free space, 
4w 10" 7 weber/amp-m. It follows from the preceding equations 

thai 

me* Nmv 1 2E k 



v — 



and 



erli NerB QnJ 



2E k 



m tB Qho2P 



(11.4) 
(11.5) 



With different gases (air, II 2 , and CO.) in the tube, Thomson 
showed that e/m had the same value 2.2 X 10" coul/kg. Thus 
cathode-ray particles are independent of the nature of the gas. 
4. Values of e/m can be obtained by a different method for 
comparison with the foregoing results. In the tube of Fig. 11.3, 
a magnetic field is established, into the plane of the paper, by 
two coils whose diameters are etjual to the length of the capacitor 
plates. The crossed electric and magnetic fields are adjusted to 
give the cathode-ray zero deflection. The cancellation of the 
electric force (upward) and the magnetic force (downward) is 
expressed by 



isV — Bicv 



(11.0) 



where A'i is the electric field intensity. 

Next the particles are deflected by a magnetic field only, 
directed perpendicular to their velocity, and the radius r of the 
path is determined from observed deflection on the screen. Here 



— = Btfo 
r 



(11.7) 



From Eqs. (11.0) and (1 1.7) 
e v Ei 



m 



v 
,B, 



rBJi-, 



The Electron 143 



(11.8) 



With this method, and using air, C0 2 and H 2 , Thomson obtained 
similar values for e/m. Again, the nature of the gas did not 
influence e/m. 



11.2 THE CATHODE -RAY TUBE 

We digress to point out the relationship between Thomson's 
apparatus and the modern cathode-ray tube. If the potential to 
be observed is applied to plates !\ (Fig. 11.4) and a potential 



-o 



p. 



(a) 




Fig. 11.4 (o) Cathode ray tube, (b) Test and sweep 
potentials. 

that increases linearly with time is applied to the horizontally 
deflecting plates /■*., the electron beam (cathode ray) traces out 
a wave form. If a saw-tooth wave form is repeated at the same 
frequency as the alternating potential on P t , the trace oi\ the 
screen is repeated each cycle and appeal's to be stationary. 



144 Looking In: Atomic and Nuclear Physics 

11.3 ELECTROLYSIS AND THE ELECTRON 

It has l>een seen (Chap. 8) that the combining properties of the 
elements can l>o interpreted in terms of an atomic theory of 
matter. Faraday's study of the electrolysis of aqueous solutions 
of chemical compounds suggested thai electricity is also "atomic" 
in nature. Faraday's discoveries of fundamental importance may 
be expressed thus: 

1. The mass of a substance separated in electrolysis is propor- 
tional to the quantity Q of the electricity that passes. 

2. The mass ;!/ of a substance deposited is proportional to the 
chemical equivalent k of the ion, that is, to the atomic mass A 
of the ion divided by its valence t>, 



M-kQ-^Q 



(11.9) 



where /"' is a constant of proportionality known as Faraday's 
constant. 

Careful measurements have been made of the amount of 
electricity required to lilierate a mass of any substance numeri- 
cally equal to its chemical equivalent, say 107.88 kg of silver, 
1 .008 kg of hydrogen, or 05.38/2 kg of ainc. This value is 9.052 X 
10 7 coul/kmole. It is called 1 faraday. It is represented by F in 
Eq. (11.9). 

Faraday inferred from his experiments that the same definite 
amount of electricity is associated in the process of electrolysis 
with one atom of each of these substances. He considered that this 
charge is carried by the atom, or in some eases by a group of 
atoms, and he called the atom or group of atoms with its charge 
an ion. In 1874, Stoney stated the hypothesis that "nature pre- 
sents us with a single definite quantity of electricity." He sug- 
gest ed the name electron for this quantity and calculated its value 
from the faraday and from Avogadro's number N*. In terms of 
values now accepted, 

/.' = jy,«e . ij,cjf)2 x 10' coul/kmole 

9.052 X 10' coul /kmole _ -„ . 

e ~ 0.0219 X m molecules/kmoie , ' 6021 X l ° C ° Ui 









The Electron 145 

11.4 CHARGE OF AN ELECTRON, MtLLIKAN'S EXPERIMENT 

Millikan, about 1909, devised a highly precise experiment based 
on the fact that electrically charged droplets of oil can be held 
stationary between the horizontal plates of a capacitor by adjust- 
ing the voltage between the plates so that the weight of the drop 
is balanced by the force due to the electric field. The "oil drop" 
experiment can be used (1) to show that electric charge occurs in 
multiples of a discrete amount and (2) to measure the value of the 
smallest charge, the electron. See Fig. 1 1.5, page 147. 

An oil drop will fall with accelerated motion until the drag due 
to air viscosity becomes great enough to balance the weight of 
the drop. For the small speeds which occur in this experiment, 
the frictioual drag is proportional to the speed of the drop. Setting 
the weight mg equal to the f fictional drag fct>) gives us an expres- 
sion for the terminal speed vi reached by the drop falling 

mg = h'i (11.10) 

To evaluate k, we take from hydrodynamics Stoke'slaw/.' = t'wnjr, 
where ij is the viscosity of air and r is the radius of the drop, 
assumed spherical. 

If the oil drop has a charge c/, due to an excess (or deficiency) 
of electrons, and if a uniform electric field is now established 
between the capacitor plates, the electric force on the drop is 



F-ft-J, 



(li.ll) 



When the drop now falls under the influence of gravitational and 
electric fields, it attains a new terminal velocity v^: 

F — mg = ft»j O'o up taken as positive) (11.12) 

Combining Kqs. (11.11) and (11.12) gives 

V 

--,<} — mg = hit (11.13) 

Suppose now that, owing to random ionization, the charge on 
the drop increases by an amount q„. There will be no significant 
change in mass, but a new terminal speed v 3 will result from the 
change in electrical force: 



y 

-j (<1 + 9») — mg = Icv-s 



(11.14) 



146 Looking In: Atomic and Nuclear Physics 

From Eqs. (11.13) and (11.14), we have a measure of the change 
in charge in terms of observed speeds: 



Qn = y &(->» - *>j) 



(11.15) 



Millikan observed that experimental values for q„ were always 
whole-number multiples of a certain quantity. Pie inferred that 
this quantity is the basic unit of charge, the electron. 

Example. By timing its full through a known distance, an experimenter 
determines "the successive -prods y„ and c„ +1 of a single oil drop having 
successive different random charges. He computes the change in charge 
(q„) from Kq. (11.15) as tabulated below. What do these data indicate 
for the charge of an electron? 

By inspection, one notes that the values for </„ are, within experi- 
mental uncertainty, whole-number multiples of 1.6 X 10"". By divid- 
ing this number into the charges q„, we find the values n in the second 
column. The experimental value for the electronic charge is (hen the 
average of the values of e in the last column. 



q,„ X10-" 


coul 


1.6 X 10~" 


n 


qn 
e = — » 
n 


X 10 "coul 


4.76 




2.98 


3 




1.59 


3.21 




2.00 


2 




1.61 


4.96 




3.10 


3 




1.65 


8.07 




5.05 


5 


e 


1.61 
= 1.61 



11.5 ELECTRONS IN METALS 

The conduction of electricity through metals is fundamentally 
different from electrolytic conduction. When a copper wire has 
carried an electric current, even for a long time, no chemical 
change can be detected in the copper. .More than a century ago 
it was first assumed that electricity is an agent that can flow 
freely in a metal. The sign of this charge and its direction of How 
were unknown, but the flow was assumed to be from the arbi- 
trarily defined positive (high-potential) terminal to the negative 
(low-potential) terminal. At present we have evidence that elec- 
trons flowing in the opposite direction are responsible for the con- 
duction of electricity in metals. These electrons are called free 
electrons because they are temporarily detached from atoms. 
The number and freedom of motion of these electrons determine 



The Electron 147 

the properties of the material as a conductor of electricity (and 
heat). 

Xo other property of solids has such an enormous range of 
values as does electrical conductivity (Fig. 11.6). The best con- 
ductors have electrical conductivities greater than that of the 
best insulators by a factor of 10". 




^=feo 



v Telescope 



Fig. 11.5 Millikan's apparatus for determining the charge of an electron. 



Conduction electrons have been compared to a gas which is 
free to move within the metal, under the influence of an applied 
electric field. The metal is visualized as consisting of an assembly 
of stationary positive ions permeated by an electron gas which 
makes the metal as a whole neutral. This qualitatively attractive 
picture proves to be inadequate in several important respects. 
first, on this basis we should expect the specific heat of a metal 
to consist of two parts: that of the ionized atoms considered as 
vibrators (3/2) and that of the electron gas (}!£), or a total 
specific heat of f ft. This finding is in marked disagreement with 



10 _ 



10" 



10° 



io J 



10 6 



10' 



to" 



io ,: 



Ohm-cm 
10" 10* 



=f 



Cu Ni Ho 



Si 
Ge 



n i i i r 

Se CU2O 8 Celluloid Mica 



ZnO 



Glass 



Amber Paraffin 
Porcelain 



Quortz Ceraml 



-Semiconductors - 



■ Insulators ■ 



■-*- Conductors™ 

Fig, 11,6 Resistivities (ohm-centimeters) vary over the enormous range of 
aboyt lO. 2 " 



Looking In: Atomic and Nuclear Physics 



148 

measured specific heat, of dements in the solid state, namely, 
5S cal . P ( mole degree) = 3/f . a relation discern! by Du hmg 
and Petit as early as 1810. Second, the variations of e tec tncal 
conductivity with temperature and the escape of electrons 
through a metal surface at high temperature (thermion.c emis- 
sion) cannot be explained quantitatively on a classical e c ct on 
gas theory, neither can the great variation m conductivity. 
Quantum theory (Chap. 16) provides a satisfactory way out of 
these difficulties. 

QUESTIONS AND PROBLEMS 

1 (a) What Ls the kinetic enemy of at, electron which moves in a 
oirce" of radius 5.2 em perpendicular to a magnetic mi uction of 9 .1 
TlO^weWom*? (6) 523 potential difie^nce would be reqmred to 
.top this electron after i, ^fctff'fi^ ,j X l0 ' volts 

2 Vn electron of mas « and charge B falls through a potential 
difference 1' and .hen enters at right angles, a region of uniform mag- 
fiS*. 8. What is the radius of the electron pa* m^mag- 

" L,t 3 Etetrom traveling 2.00 X 10' m/see are subjected to a magnetic 
i,uha-tb f 0.0030 weber/m» in the apparatus „^ by Thomson 
£K3> to Snee/Ma) The capacitor plate '■^««J"J- 
wSt voltage imiM be applied to them to return the beam to its m- 
J ^teJ 1 StioBl (») SL in a .ketch the ^*£~«*» 
eleclrnu velocity r. the maguel.c induction ft and the Uu tnUuld fc.^ 

4. An oil droplet of mass 2.5 X 10"" gm. which <^ "MH^d 
2 electron charges, is between two horizontal capacitor p ate* 2 em 
JpSZnS ft* *e droptef is entirely.snpported by etectne foraefe 

from falling, what must be the potential ddTerenrc holi v^^Pjg 

10 rft *SKb- photograph, an ejeotron pato* bjrfWo 

a circle of 12-cm radius bv a magnetic induction of O.OOsO weber/m^ 
?«i<W»lutP the enemy of the electron (*) Calcdate tho enemy of an 
Hec.ron whose path radius is 20 cm in the *.,,,, {«• d- 

/Ins. 3.6 X 10-' 6 joule or 2.2 Mev, ^.U X iw J"""- 

6 V narrow beam of electrons traveling with speed e along toe 

x & t pa s^s etwecn the horizontal plates of a parallel-plate eapacrig 

F ,3 Htamed lo a potential difference V. The separation of the 

plan-s 111 Show that in traveling a distance ( in the capuclor the beam 

will experience a deflection from J lie x axis, given by 

„ t V I 

9 = - T ~\ 



12 

Ions 

and Isotopes 



The most important trends in indus- 
try today spring from an increasing 
knowledge of the properties of atoms 
and their component parts. 

David Sarnoff 



The discovery of isotopes was foreshadowed by studies in radio- 
activity (Chap. 17) about 1007. The possibility that two different 
radioelements might lie identical chemically was inferred by the 
failure to separate certain ones by any chemical means available. 
Also Thomson's study of positive rays (HJl.Tj in ion-deflection 
experiments yielded two lines for neon (atomic masses 20 and 22) : 
yet no dilTereuees were observed in the optical spectrum of the 
Ne gas. Thomson recorded his suspicion that "the two gases, 
all hough of different atomic weights, may be indistinguishable 
in their chemical and .spectroscopic properties." 

12.1 MASS SPECTROMETERS 

Boon after World War I, Dempster, Aston, Bainhridge, and others 
devised instruments for determining both the masses of isotopes 

149 



150 Looking In: Atomic and Nuclear Physics 

and their relative abundance. Although there are many types of 
mass spectrometers,* a brief consideration of their common ele- 
ments should clarify the operation of any type. 

A moving particle might be characterized by its velocity v, 
its momentum mv, or its kinetic energy \rns-. We may consider 
arrangements of electric and magnetic lie Ids designed to sort 
charged particles according to these properties. 

Two types of energy selector are suggested in Fig. 12.1. In 
the first, ions from a source 8 are accelerated through a potential 



i.ri 

J""\ 

{a) 
Fig. 12.1 Energy selectors. 




(6) — :— 



difference V. They acquire kinetic energy ^mu 2 equal to Vq. The 
speed of an emerging ion is then 



\ m 



(12.1) 



We have tacitly assumed that the ions are at rest at s. Actually 
they may have small (I ev) energies of thermal motion. This is 
usually negligible compared with the energy {say, >100cv) ob- 
'ained from the electric field. But for some purposes, the slight 
spread in the velocity values for emerging particles might have, 
to lx> considered. In Fig. 12.16, the ion beam passes between the 
plates of a curved capacitor. The ions are acted upon by an elec- 
tric field /■-' in the direction of O. The ions move in a circular arc. 
The centripetal force is provided by the electric field: 



mv* r. 
- W = Eq 



(12.2) 



* in a mass spectrometer, a meter measures an ion current; in a mass 
Spectrograph, the record is obtained on a photographic plate. 






Ions and Isotopes 151 

This equation can be rearranged to show that the device is an 
energy selector 



&nv* = %REq 



(12.3) 



In the selector of Fig. 1 2.2, the beam of positive ions passes 
through a region where a magnetic induction B is directed out- 



Fig. 12.2 


Momentum selector (B is directed out; 


/V'.lJ 


E = 0). 




,' -A. R 
r < * v « 
bid N o 



ward from the page. By equating the centripetal force to the mag- 
netic side thrust, we get 



mi- 
li 



= qvB 



(12.4) 



and it is apparent that this tie vice is a momentum selector 
mv = RBq (12,5) 

Consider next a twain of positive ions acted on by an electric 
field li and a magnetic induction B at right angles to each other. 
In the situation of Fig. 12.3, the ions experience an upward force 



B„„, 



Fig. 12.3 Velocity selector. 



1 + 

— *■ 

I 



fr. 



I 



qE due to the electric field. With B directed out from the page, 1 he- 
magnetic side thrust on the ions is qvB, downward. Only those 
ions for which 



qii = qvB 



(I2.fi; 



will pass through without any deviation. This filter selects ions 
of a particular velocity 



E 
S = B 



(12.7) 



152 Looking In: Atomic and Nuclear Physics 

I» a Baiubridge mass .spectrograph (Fig. 12.4), positive ions 
from an ion source arc colli mated by slits s, and x* and then pass 
into a region in which they experience an electric force to the left 
and a magnetic force to the right (supplied by Hi which is directed 
into the page). From Etj. (12.7), the only ions which pass through 
slit « 3 are those of speed 



r = - Bi 



(12.8) 



where /; has Iktu expressed as V ".-,-. Beyond s ;i the ion is influenced 
only by a uniform induction B» (into the page) which causes the 




Fig. 12.4 Boinbridge moss speetro- 
groph. 



ion to move in a circular arc of radius ft until it strikes the photo- 
graphic plate /' where it makes a developable trace. By measuring 
the distance 2/1 from this trace to slit s a , and using Eq. (12.4), 
we can find the charge mass ratio 



'/ _„ '' 
m BJl 



V 



B y B#R 



(12.9) 



The charge on the ion will he a multiple of the charge on the, 
electron (usually U or 2c), which the experimenter must find. 
With q known, he can calculate the mass m of the ion. lie may 
then add the mass of the missing elect ron(s) to (hid the mass of 
the atom. 

For measurements of highest precision (a few parts in I0 7 }, a 
mass spectrometer is designed to cover only a limited region in 



Ions and Isotopes 153 

the mass scale from I to 2">0. Unknown masses are found by inter- 
polation between known masses (often using ions of molecules as 
well as of atoms). Also, atomic masses can be deduced from the 
energy release in certain kinds of nuclear reactions (Chap. 18). 
A table of the "Ix-st" values of atomic masses obtained as aver- 
ages of mass spectrometer and nuclear reaction data, adjusted 
for self-consistency, is useful in many calculations in modern 
physics. * 

12.2 ISOTOPES 

As the accuracy in measurement of atomic masses increased, it 
was established that not. all atoms of the same element have the 
same mass. Atoms of the same element (same Z) which have dif- 
ferent masses are called isotopes. Many elements (Be, F, Xa, Al, 
P, Co, etc.) occur naturally with only one isotope. Many others 
(H, He, Li, B, etc.) have two, and tin, the most varied, has no 
less than 10 isotopes. 

It the mass of a carbon atom is taken to be exactly 12, then 
the masses of the other elements, determined by quantitative 
chemical analysis, come out to be nearly whole numbers. Histori- 
cally this led to Front's hypothesis that all elements were built 
from hydrogen. This picture was spoiled by certain atomic masses 
determined chemically: 35.8 for CI, 63.54 for Cu, etc. But when 
measurement of isotope masses became possible, it was found 
that the mass of every isotope of every element was very close in 
an integer on the scale in which carbon is taken as 12 antra (or, 
originally, oxygen defined as 10 amu). Naturally occurring chlo- 
rine, for example, is a mixture of about 7o per cent of an isotope 
:S4.!)7!K) and 25 per cent of an isotope 38.9773. Its average mass, 
as found in chemical experiments, is then §(35) + t($~) = 35.5. 

The whole-number rule may be retained in this form: The 
mass of every isotope of every element is well within 1 per cent of 
a whole number when expressed in atomic mass units, defined by 
taking carbon as 12 amu, exactly. We thus retain the picture of 
all atoms built up of some unit of which there is 1 in hydrogen, 
4 in helium, Hi in oxygen, etc. We have yet to explain, however, 

* Sec (.'. II, Blunt-hard, C. R. Burnett, H. G. Stoncr. and R. L Weber, 
"Introduction to Modern Physios," appendix fi, pp. 392 100, l'rciitire- 
Hall, liic, Engleivood Cliffs, X. J. , 1958. 



154 Looking In: Atomic and Nuclear Physics 

why the atomic masses arc not exactly integers instead of being 
very nearly integers. 



QUESTIONS AND PROBLEMS 

1. The values of E and B in the velocity selector of a mass spec- 
trometer are MIDI) colts, n. and 0,050 weber/m*. (fl) Wluit will be the 
speed of ions passing through this selector? (b) By what radius will a 
singly riiarsnl ion of mass 50 amu be deflected by a magnetic field of 
2.5 X 10~ a weber/m 1 after leaving the velocity selector? 

Arts. 1.6 X 10' m/aee;2.1 em 

2. A singly charged positive ion is accelerated through a potential 
difference of 1,000 volts, li is then subjected to a magnetic field of 
0.10 weber/m 1 in which it is deflected into a circular path of radius 
18.2 cm. (a) What is the mass of the ion? (h) What is the mass number 
of the ion? Am. 2.05 X 10 ■'■ kg. 15.9S amu: If. 

3. A dust particle has a mass of 3.0 X 10~ s kg and a charge of 
5.0 X 10 10 coul. The particle is accelerated in an electric held until it 
has a speed of 4.0 m/sec. (a) Calculate its kinetic energy in joules, (b) 
What potential difference is required to give the particle this speed? 
(c) If the particle moves at right angles to a magnetic induction of 
0.20 weber/m 2 , what force will the particle experience? (</) What is the 
radius of the circular path in which the particle will move in this mag- 
netic field? 

Ans. 2.4 X 10"' joule; 480 volts; 4.0 X lO" 10 newtnn; 1.2 X 10 a m 

4. In a Dempster mass spectrometer, positive ions formed by 
heating a salt of an element are accelerated to a slit s, by a potential 
difference V (about 1,000 vol Is). A narrow bundle of ions then passes 



Photo, plote or 
electronic detector 



Fig. 12.5 




through the slit into a semicircular chamber where there is a magnetic 
induction II perpendicular to the ion velocity i> (Fig. 12.5). Ions having 
different values of e,'m will travel arcs or different radius. Show that the 
charge/mass quotient of an ion can be computed from 



e 
m 



2F 
fi ! r 2 



Ions and Isotopes 155 

5. On the photographic plate of a ma>s spectrograph, a trace made 
by a singly charged ion is h.und jusf halfway between the line formed 
by 16 (+) and that formed by CH,(+). Find (he mass of this ion if the 
mass sped rennet er is (u) a Bainbridge type, when 1 the mass of an ion is 
proportional In the radius m = kr, and (b) a Dempster type, where the 
dispersion equation is »i = kr 1 . 

Ans. 10.018148 amu; 1B.009074 amu 

6. In a method devised by S. A. Goudsmit, masses of heavy ions 
are determined by timing (heir period of circulation in a known mag- 
netic field. To get an idea of the timing requirements, calculate the 
period of revolution of a singly charged ion of iodine ail 127 (mass 
126.945 amu) in an induction of 0.045 weber/m 5 . 

-4ns, about 1.8 X I0~* sec 

7. Show why the mass spectograph gives data on the atomic masses 
of individual ions, while conventional chemical methods yield results 
only on average atomic, masses. 

8. Silicon has an atomic number of II. Consider two isotopes of 
silicon having mass numbers 28 and 30, fill in ihe remaining spaces in 
the table: 



Mass number. , 28 30 

Number of electrons in the atom 

Number of positive charges in the nucleus 

Number of protons in the nucleus 

Number of neutrons in the nucleus. 



13 

Electromagnetic 
Radiation 



Electricity, carrier of light and power, 
devourer of time and space, bearer of 
human speech over land and sea, greatest 
servant of man. Charles Eliot 



I. CLASSICAL THEORY 

In 1 8(14 , James Clerk Maxwell completed the structure of classi- 
cal electric and magnetic theory. His summarizing equations 
stand with Newton's laws of motion and the- laws of thermody- 
namics as masterpieces of intellectual achievement. The four dif- 
ferential equations show how electric and magnetic fields are 
related to the charges and currents present and how - they are 
related to each other. They correlated experiments in large areas 
of physics and predicted important new results. Specifically, 
Maxwell showed that a changing current will radiate electromag- 
netic waves in which E and li are perpendicular to each other and 
to the direction of the wave motion. Tlis theory predicted that 
electromagnetic waves of all frequencies should travel with tiie 
speed of light c, whose numerical value can be determined by 

156 







Electromagnetic Radiation 157 

measuring the force between currents. This theory was experi- 
mentally verified by Hertz in 1888, and by 1901 Marconi suc- 
ceeded in transmitting electromagnet ic signals across the Atlantic 
Ocean. The. electric generator, motor, betatron, television, and 
radar are based on principles included in Maxwell's equations. 
Home of the relations and experimental facts which Maxwell 
synthesized carry the names of earlier investigators,* 

13.T GAUSS' LAW FOR AN ELECTRIC FIELD 

Imagine a potatolike surface immersed in an electric field. The 
flux *« of the electric field through this arbitrary surface is 
measured by the number of lines of electric force that cut through 



Fig. 13.1 
surface. 



Electric flux through o 




the surface. Let the surface be divided into elementary squares 
An small enough so they may be considered to be plane. An cle- 
ment of area can be represented by a vector As whose magnitude 
is proportional to the area: the direction of .is is taken as the 
outwarddrawn normal to the surface element. 

The field intensity E is practically constant over an element- 
As; B is the angle between K and As. The flux is found by adding 
up the scalar product A* ■ As cos 8 for all the elements into which 
the surface has been divided: 

*k = 2E-As (13.1) 

For a more precise definition we replace the sum by an integral: 

* K - /E • r/s (13.2) 

The integration is to be taken over a closed surface. 

* A reader not familiar with calculus notation may wish (o skhn 
Sees. 13.1 to 13.4 and resume his study immediately after the statement, 
of Maxwell's equations in Sec. 13.5. 



158 Looking In: Atomic and Nuclear Physics 

Gauss' law states that the net (outward) electric flux through 
any closed surface is equal to l/*« times the net charge q enclosed 
hy the surface : 

eu $ K = q or «,,/£• ds = q (13.3) 

Gauss' law provides a convenient way of calculating E if the 
charge distribution is symmetrical enough so we can easily evalu- 
ate the integral in F.q. (13.3). 

Example . A long copper tube of radius a has ;i charge of +q/l per 
unit length. It is surrounded by a coaxial copper tube of radius !> which 
earrics charge per unit length — q/l. Find the electric field (a) at a di* 
tanee ri from the axis where 6 > r t > a and (l>) outside this coaxial 
cable at, distance r : from the axis, Fig. 13.2. 




Fig. 13.2 






in) Draw a gaussian surface which is a cylinder of radius r u coaxial 
with ihe cable. Since the electric field is radial, there will be no flux 
through the cutis of the cylinder. For a length I of cable Eq. (13.3) 
becomes 

*u/E • da = t E(2irri)l - q 
giving 

(6) When the gaussian surface is a cylinder of radius r-i, the net 
charge within it is (+q/l - q/l) = 0, and hence E = outside the 
outer conductor. 

13.2 FARADAY'S LAW OF INDUCTION 

A changing magnetic field produces an electric field, as described 
in Faraday's law (Sec. 10.12). Consider a test charge (/ which 



Electromagnetic Radiation 159 

moves around the circle in Fig. 13.3, where a uniform magnetic 
induction B is directed out of the paper and is increasing with 
time. The work done on the charge, per revolution, is the product 
of the emf S and the charge q. The work is also the product of the 



Fig. 13.3 CKorged particle q moving in a 
magnetic Induction 6. 




force qE that acts on the charge and the distance 2irr. Equating 
these two expressions gives 

£ = 2iffli (13.4) 

or, more generally, 

8 = 6 E • dl (13.5) 

If this last equation is combined with £ = —d$ B /dl, Faraday's 
law of induction can be written 



(13.G) 



13.3 GAUSS' LAW FOR MAGNETIC FIELDS 



Gauss' law for a magnetic field expresses the fact that in magne- 
tism there is no counterpart to the free charge q in electricity. 
Isolated magnetic poles do not exist. Hence the magnetic flux 
'l>; ( through any closed surface must be zero: 



*b = <f> B • da = 



13.4 AMPERE'S LAW 



(13.7) 



Ampere's law (Sec. 10.10) giving the relationship between current 
i and magnetic induction li can be written in circuital form as 



- <6 B ■ dl - i 

Ho T 



(13.8) 



160 Looking In: Atomic and Nuclear Physics 

The line integral can be applied to any closed path near the cur- 
rent; symmetry usually suggests the most convenient path. 

Example. Find the magnetic induction Si al a distance r from a long 
Straight wire currying current i. 



Fig, 13.4 Magnetic induction S near a 
long, straight wire. 



Consider a circle of radios r centered af I lie wire for t he path of 
integration. Since It is tangent i<> the circle al each point .4, vectors It 
ami <l\ (the element of are) poinl in (lie same direction. From symmetry, 
It has tiie same magnitude at each point on the wire. Equation (138) 
becomes 




- 6(27rr) = ir 



B = **- 

2irr 



winch is the result obtained in Sec. 10.10. 



Experiments show that just as a changing magnetic held 
induces an electric field (Faraday's law, Sec. 10.12), a changing 
electric field induces a magnetic field. Faraday's law for an in- 
duced emf G may be written 



i 



E 



dt 



(13.9) 



The analogous expression for the magnetic induction produced 
by a w hang ing electric field is 



- <p II • (II = eg fjl 



/"'■ 



(13.10) 



where the constants n» and a, are required in the mks system of 
units we are using. The situation expressed in Fq. (13.10) can be 
visualized by considering the region between the plates of a 
capacitor (Fig. 13.">a) which is being charged with a steady cur- 
rent i. The accompanying dl'I/dt produces a magnetic field: B is 
shown For four arbitrary points in Fig. 13.56. 



Electromagnetic Radiation 161 

In considering the two ways of setting up a magnetic field, 
(I) by a changing electric field dK dt and (2) by a current i, we 
have assumed that there is uo current in the space considered in 
(1) and that no changing electric fields are present in (2). But, 



(a) 



(b) 




Fig, 13.5 The charging of a capacitor (a) produces a 
changing field dE dt, which (b) produces a magnetic field 8. 

in general, the contributions of both dE/dl and i must be con- 
sidered. Maxwell generalized Ampere's law, writing it in the form 



££«.«- 



dt 



(13.11) 



The term t,\{d$y.'dl) has J he dimensions of current and is often 
called the diaplawmvtit mrri-nt. Tims, although the conduction 
current i is not continuous across the gap of a capacitor (because 
no charge is transported across the gap), there is a displacement 
current % D in the gap equal to the external i. 

13.5 MAXWELL'S EQUATIONS AND ELECTROMAGNETIC WAVES 
We assemble here the four basic equations we have just discussed : 



*/ f = 



0E-rfl= ~ d 



(1 3.3) 
(13.7) 

(13.9) 

[ - <£ n ■ di = t^-%? + i (i3.li) 

m<> / dt ' 

We have written .Maxwell's equations in the form they have when 



I 









162 Looking In: Atomic and Nuclear Physics 

no dielectric or magnetic material is present, since we are chiefly 
interested in radiation in "free space." 

Consider that the start of an electromagnetic wave occurs at 
the termination of a transmission line whicli is energized by an 
oscillating electric circuit. Figure 13.fi suggests how the electric 
field lines break away from an electric dipole as the charges 
+q and —q first approach and then recede from each other in 
successive time intervals of one-eighth of a period 7. An observer 
at /' looking toward the antenna will "see" an instantaneous 
electromagnetic field pattern with E down and B toward his right 






t = 



r/a 



r/4 



3T/3 




Fig. 13.6 Radiation from an oscillating electric dipole. [Adapted from S. S. 
A ft wood, "Efecfric and Magnetic Fields," John Wiley & Sans, Inc., New York, 
1949.) 



(into this page). A moment later, as the wave advances, these 
directions will reverse. 

If an observer is at a relatively large distance from the source, 
the waves that move past hi in will be practically plane waves. 
For the waves shown in Fig. 13.7, 

B = B« sin (kx - wt) (13.12) 

E = Eo sin (kx - at) (13.13) 

Consider the rectangular prism I • I ■ dx to be fixed in space. Its 
trace in the z.r plane is shown in Fig. 13.86. As the wave passes 
over it, the magnetic flux <t?n through the rectangle will change, 
inducing electric fields around the rectangle. The line integral of 
Eq. (13.9) is dE h ; there is no contribution from the top or bottom 
of the rectangular path where dl and E are perpendicular. The 






Electromagnetic Radiation 163 

magnetic flux *« for the rectangle is B(tdx). By differentiation 

this gives 

d$n , , dB 

W = ldx -dl 

From Eq. (13.9) we have dE I = -Idx (dB/dt), or 



dE 
dx 



dB 

dt 



(13.14) 



where we have changed to the d notation of partial derivatives to 
indicate that both B and E are functions of x and ( but that in 




n c y 



E-r<fE 



N 



i 



B+rfB 



dx 
(c) 



Fig. 13.7 to) Section of a plane electromagnetic wave troveltng to the right 
E parallel to i axis, 6 parallel to y axis. The part marked dx is viewed in (b) 
in the xz plane, and in (c) in the yx plane, 

evaluating dE/dx, t is assumed constant (Fig. 13.7// is an instan- 
taneous "snapshot"). Also, x is assumed constant in evaluating 
dB/dl at the particular strip in Fig. 13.7c. From Eqs. (13.12) 
and (13.13), 

kE cos (kx — oil) = o>Bn cos (kx — o>t) 

and we have the relation 



E _ CD _ 

F ~ & ~ c 

where 

angular frequency u> = 2ir/ 
wave number k = 2*/X 
speed c = f\ = a>/k 



(13.15) 






164 Looking In: Atomic and Nuclear Physics 

The ratio of the amplitudes of the electric and magnetic com- 
ponents is equal to the speed c of the electromagnetic wave. 

In considering the trace of our rectangular prism in the yx 
plane, Fig. 13.7r, we see that as the wave moves by, *« changes 
with time and a magnetic field is induced at each point around 
I d.r. This induced H is the magnetic component of the electro- 
magnetic wave; K and li each depend on the time rate of change 
of the other. Since there is no conduction current £ in the space 
considered, the line integral of Bq. (13.10) hecomes —iilli. The 
flux *s through the rectangular element Idx is E{ld.t). By dif- 
ferentiation this gives 

et*a , , dE 

— tt- = l ax -jt 

M dt 



From Eq. (13.10) 



an 



dE 



aJ"** ,0 "« 



(13.16) 



where partial derivatives have been indicated for the same reasons 
as in Eq. (13.14). From Eqs. (13.12) and (13.13) 

hllu COS {kx - ut) = — enflqfi'o COS (kx — <>)t) 

and since c = w/k, we have 

U = _L = l (13.17) 

liu e<ijuiiw tviinC 

By eliminating the fields between Eqs. (13,15) and (13.17), we get 
c - — j— (13.18) 



Substitution of numerical values in this relation gives 

I 



c = 



v/(8.!) X 10 l2 eoulViit-m ! )(4jrl0 ' weber, ,'nt-m) 
- 3.0 X 10" m/sec 

which is the speed of light in free space. 

Maxwell made this calculation before it was recognized that 
light was electromagnetic in nature and before Herts had detected 
electromagnetic (radio) waves. 



Electromagnetic Radiation 165 

The magnetic part of an electromagnetic wave is often de- 
scribed by the magnetic field strength //, rather than by the mag- 
netic induction B. For waves propagated in free space, the only 
case we shall consider, 



H = -B 



(13.19 



Since ju« = 4r X 10 -7 weber/amp-m. // has the dimensions 
(weber m'-'i (amp-m/ weber), or amp/ in. The quantity EH = 
vVu/tu = 376.7 ohms is sometimes called the resistance of free 
space. 



13.6 ENERGY FLOW: INTENSITY OF A WAVE 

Consideration of a parallel-plate capacitor can lead us to an 
expression of genera! validity for the* energy stored in an electric 
field. To charge the capacitor to a potential difference A V requires 
work W equal to AC AV*, where C is the capacitance. This energy 
is stored in the capacitor a.s potential energy. Between the plates 
there is an electric field E = AV/s, where « is the plate separa- 
tion. The capacitance is given in terms of the area A of the plates 
and the permittivity e u as C - t„A/s. The stored energy can then 
be expressed as 

W = £C(AF) 2 = i*,- EV = U E*As 

Since .4s is the volume of the space between the plates of the 
capacitor, we may associate the energy with the field, defining 
the energy density of the electric field as the energy per unit 
volume: 

u, = (k) (u E* (13.20) 

Similarly, consideration of the work required to establish a cur- 
rent in an inductive circuit suggests that the energy density for 
a magnetic field is 

„„. - ;„„//* - |B* (13.21) 

In a region where both electric and magnetic fields exist, the 
energy density is given by 

w = U lt E- + -^H* (13.22) 



166 Looking In: Atomic and Nuclear Physics 

The rale at which energy is transported by a wave across unit 
area per unit time is called the intensity .S' of the wave. Since the 
wave speed is c, 



S - c(i«P + innH*) 



(13.23) 



By the use of Eqs. (13.17) to (13.19), the energy density can be 
written 



w = s/tofie EH = ~ EH 
c 



and the equation for the wave intensity becomes 



S = c- EH = EH 
c 



(I3.1M) 



(13.25) 



The directions of E, H, and the velocity of the wave are properly 
related if we define the intensity S as the vector cross product 



S= ExH 



(13.26) 



This vector is called Poynting's vector. In mks units, it gives the 
intensity in watts per square meter. Although we have derived 
this expression for the special case of a plane wave, it is a general 
relation which can be derived from Maxwell's equations. 

Example. Consider a plane monochromatic plane-polarized electro- 
magnetic wave (raveling horizontally northward, polarized vertically 
(eleeirie field intensity directed alternately up and down). The frc- 



y (up) i 1 



x (north) 




Fig. 13.8 



Z (east) 



quency is 5.0 megacycles/see. The amplitude (maximum value) of the 
electric field intensity E is 0/160 volt/m. (a) Give an analytical ex- 
pression for this wave, (6) Find the average intensity of this wave. 



Electromagnetic Radiation 167 

The wavelength is X = c/f = 60 m. The period T = 1// = 2jt/« 
= 2.0 X 10 -7 sec. If coordinates are chosen as in Fig. 13.8, from Eq. 
(13.12) the electric field is 



B, = B. - 

E v = E B sin 2jr 



(H)" 



060 sin (irl0 7 f - 0.105s) 



The amplitude //„ of the magnetic field is 

//„ = . p Bo = 0.060 volt/m — — , - = 1.59 X 10 < amp/m 
Vj«o 37 G./ ohm 

Since the magnet ie field is confined to the east-west direction, 
//, =0 //„ = //.- = 1.59 X 10-* sin (jtIO'J - 0.105s) 

The Poynting vector S = E X H is in the x (north) direction, and its 
magnitude is 



5 = #o//u sin s 2v 



(*-*)-■ 



54 X 10-«sin= (xlO' - O.lOox) 



The average value of sin- o over a cycle is .V, so the average intensity 
of this wave is 

S = hEoIh = 4.74 X 10-» \vatt/m» 

Maxwell's electromagnetic theory explained the then-known 
properties of light ; the experimentally measured speed of light in 
free space, polarization, interference, and diffraction, and the 
dispersion that occurs when light, passes through a medium where 
wave speed depends on wavelength. Extended to x rays, the 
theory identified them as also electromagnetic radiation, the dif- 
fraction of x rays by a crystal lattice being similar to the diffrac- 
tion of light by a ruled grating. Many seemingly diverse radia- 
tions were shown to be related regions of an electromagnetic 
spectrum of grand extent — some 80 "octaves," of which the 
visible spectrum comprises a little less than one octave. 



II. QUANTUM EFFECTS 

Beginning in 1900, developments took place which indicated that 
Maxwell's theory does not predict accurately all aspects of elec- 
tromagnetic radiation and absorption of energy. These develop- 
ments led to the quantum theory. We shall trace the quantum 



168 Looking In: Atomic and Nuclear Physics 

hypothesis from its origin in explaining blackbody radiation, its 
confirming success in explaining the photoelectric effect and 
Compton effect, and its striking hut limited success in the Bohr 
model of the atom (Chap. l;">) to its merging with other hypothe- 
ses in wave mechanics (Chap. 16). 



13.7 BLACKSODY RADIATION 

Any object continually emits and absorbs radiation, exchanging 
energy with its surroundings. If the temperature of the object is 
high enough, the radiation may be seen- the material glows. 
There is a direct relation between absorption and emission. 
Kirchhoff's law stales that an object which absorbs radiation of a 




Fig. 13,9 
radiators. 



Good absorbers are good 



particular wavelength strongly is also a strong radiator at that 
wavelength. Consider a platinum disk in a furnace (Kg. 13.9) at 
thermal equilibrium. Suppose that it receives one unit of energy 
per unit area per unit time and that the fraction p is reflected. 
Then I — p is the fraction absorbed, designated a. Hut if the tem- 
perature of the disk is to remain constant, the disk must lose 
as much energy per second as it receives. The rate of emission t 
from the area considered must equal a. If a carbon disk is in 
thermal equilibrium in the same furnace, it receives the same 
energy per unit area per unit time as does the platinum, but it 
absorbs a larger fraction (1 - p') and hence must emit more, 
to keep T constant. Good absorbers of radiation are good emit- 
ters, as represented by the relative lengths of the arrows for 
p, p', t, and «' in Fig. 13,9. 



Electromagnetic Radiation 169 

All materials exhibit characteristic differences in their absorp- 
tion of radiation in different parts of the spectrum. (The colors of 
many things we view are due to such selective absorption.) 
Hence the emission spectrum for thermal radiation at a given 
temperature depends on the material of the emitter. 

We can imagine an ideal body which absorbs all radiation 
incident upon it. By KirehholV's law, this body would also be 
the most effective emitter of thermal radiation at all wavelengths. 
Such an ideal absorber-emitter is culled a blackbody. 

Fortunately it is possible to realize blackbody conditions 
experimentally to any degree of approximation requited. If we 
form our material to make a cavity with a small opening to the 
outside, the hole will behave as a blackbody. Radiation which 
enters the hole will bounce around at the inner walls of the cavity, 



Fig. 13,10 Cavity radiation Is nearly 
blackbody radiation. 





gradually being absorbed. Only a tiny fraction of the radiant 
energy will be reflected back through the hole. Viewed from the 
outside, the hole is an excellent approximation to a blackbody. 

When the walls of a blackbody cavity are maintained at some 
temperature T, the interior is filled with radiation. A tiny fraction 
leaks out of the hole. A ray leaving the hole has, in general, 
undergone many reflections. At each, reflected energy is added to 
emitted energy until in the emerging (blackbody) radiation, the 
energy distribution depends only on the temperature of the cavity 
and not on the material of which its walls are made. 

The blackbody radiation can lie dispersed by a grating, and a 
bolometer or thermopile can be used to measure the energy radi- 
ated in each wavelength interval. A continuous spectrum is 
found; that is, radiation at all frequencies is observed. The dis- 
tribution of intensity per unit wavelength interval is shown in 
Fig. 13.11. When the temperature is increased, more energy is 
radiated at every frequency, and the relative increase is greater 
at the higher frequencies, shifting the maximum of the intensity 



170 Looking In: Atomic and Nuclear Physics 

distribution to lower wavelengths (Fig. 13.11a) or higher fre- 
quencies (Fig, 13, lib). 

The area under the blackbody distribution curve f " S(f) df, 

whirl] represents the total power radiated, is found i" increase as 
the fourth power of the absolute temperature: 

P-eAT* (13.27) 

where P is the radiant flux from a blackbody of surface area A 
at absolute temperature T. This is known as the Stefau-Boltz- 
mann law ; the constant a has the value 5.7 X 10" 8 watt/ (m*) (°K*). 



Intensity 



I 3000 : K 




Intensity 




2.0 4.0 

(o) Wavelength, microns 



(6) 



Frequency 



Fig. 13.11 Blackbody radiation distribution: (a) intensity vs. wavelength, with 
visible region dotted, (b) intensity vs. frequency, with Ti < T. < It. 



13.8 PLANCK'S LAW 

A number of physicists advanced theories based on classical 
physics to explain the distribution of energy in the continuous 
spectrum from a blackbody. Lord Raylcigh and Sir James Jeans 
assumed that radiation in a cavity has degrees of freedom which 
correspond to the frequencies of standing waves that are possible 
in the cavity and that the energy is divided equally among these 
different degrees of freedom. The resulting distribution law is 






(Rayleigh-Jeans) 



(1.3.28) 



where K» is the radiancy (power per unit area) at a wavelength 

X and ci and c a are empirical constants. The Rayleigh-Jeans law 
fits the experimental data (Fig. 13.12) only for large values of 



Electromagnetic Radiation 171 

XT', fn fact, it leads to an "ultraviolet catastrophe" by predicting 
that as X becomes smaller, R\ increases without limit; the total 
power radiated by any body is infinitely large! 

Wilhelm Wien assumed that cavity radiation came from 
molecular oscillators among which energy was distributed with 
respect to frequency according to a Maxwell distribution (similar 
to the distribution law successfully used for molecular speeds in 
the kinetic theory of gases). The resulting distribution law agrees 
with experimental data in the short-wavelength region of the 



Intensity 



Planck's law 




12 3 4 

\Wgvelength, microns 

Fig. 13.12 Agreement of radiation formulas with 
Coblentz's experimental data (circles). 



spectrum, but predicts values of I{\ which arc too low in the region 
where X7' is large: 



th = cV-V— '" (Wien) 



(13.29) 



Max l'lanck also started with the assumption that the wallt- 
of a cavity radiator are made up of tiny electromagnetic oscilla- 
tors or resonators of molecular dimensions. He, too, used a Max- 
well distribution, taking e~ HlkT as the probability that an oscil- 
lator has energy /■.'. lie accepted the Rayleigh-Jeans calculation 
for the number of oscillators per unit volume in the frequency 
range from / to / + df. But Planck was led to make two radical 
assumptions: 



172 Looking In: Atomic and Nuclear Physics 

1. An oscillator can have only energies given by 
E — nhv 



(13.30) 



where c = frequency 

k = Planck's constant (of "action") 
n = an integer (now called a quantum number) 
The equation asserts that the. energy of the oscillator is 
"quantized." 

2. An oscillator does nut radiate continuously (as expected on 
.Maxwell's theory) but only in quanta of energy, emitted when 
an oscillator changes from one to a lower of its quantized energy 
states. The quantum (or photon) radiated has energy propor- 
tional to the frequency of the wave: • 



K = hv 



(13.31) 



From these assumptions, Planck derived the distribution law 



R*- 



2irAc 2 



X 6 e*"* T - 1 



_CiX-« 

,,r, \r _ 



- (I'lanck) 



(13.32) 




where c = speed of light 

k = Boltzmaiui constant 

h = (Mi2"» X 10" w joule/sec 
Planck's law has been written in the second form with empirical 
constants Ci and c* for comparison with Eqs. (13.29) and (13.30). 
For a wide range of temperatures (300 to 2000° K) and a wide 
range of wavelengths (O.o to n2 ft), Planck's law represents the 
experimental data within 1 per cent. It is interesting to note that 
despite its initial success the quantum hypothesis was resisted by 
Planck himself. Conservative in nature, he tried for years to 
reconcile his "quantum of action" (A) with classical theory. 

13.9 PHOTOELECTRICITY 

bight or other electromagnetic radiation falling on the surface of 
a metal (Fig. 13.13) can under certain circumstances liberate 
electrons from the metal. The number of electrons emitted pet 
second can he determined hy measuring the photoelectric current. 
The energy distribution of the electrons can he determined by 
applying a retarding potential and increasing it gradually until 



Electromagnetic Radiation 173 

the stopping potential V, is found for which no electrons reach 
the collector. 

The chief features oT photoemissiou are : (1 ) There is no detect- 
able time la<i (> 10 ' sec'i between irradiation of an emitter and 
ejection of photoelectrons; (2) the number of electrons ejected 
per second is proportions! I to the intensity of radiation, at a given 
frequency; (3) the photoelectrons have energies ranging from 
zero up to a definite maximum, which is proportional to the fre- 
quency of the radiation and independent of its intensity; (4) for 
each material there is a threshold frequency v K below which no 
photoelectrons are ejected. 



* Light source 




Fig. 13.13 Apparatus for photoelectric effect. 

These characteristics of photoemissiou cannot be explained 
by Maxwell's theory of electromagnetic radiation. In MM)"> 
Kinslein made the assumption that light of frequency c can give 
energy to the elections in f he metal only in quanta of energy hv. 
Fit her an electron absorbs one of these quanta, or it docs not. 
If it is given energy hv, an electron may use an amount of energy 
v> in escaping from the metal, where it has negative potential 
energy, into the vacuum, where it has zero potential energy. The 
quantity w is called the work function of the surface. The maxi- 
mum kinetic energy which the electron can have when it leaves 
the surface is therefore 



/■: 



*. NIEIJS 



= hv — w 



(13.33) 



This is called Ivinstein's photoelectric equation. It explains the 
linear relationship E i: = an + h shown in Fig. 13.1-1: The slope a 
measured from the graph agrees with the value of Planck's con- 
stant A; the negative intercept b is identified with the work func- 



174 Looking In: Atomic and Nuclear Physics 

lion id of the metal. The intercept on the frequency axis is the 
minimum frequency of light that will liberate electrons from the 
particular metal. At this threshold frequency vt, the photon 
delivers just enough energy to enable the electron to get out of 
the metal (with E k = 0): 



hvo = w 



(13.34) 



From Eq. (13.33), #*.„„,* is independent of the intensity of illumi- 
nation, in agreement with experiment. 

The term photoelectricity includes several distinct phenomena. 
In the external photoelectric effect (photoemission), electrons 



*■!, M. 




Fig. 13.14 Dependence of maximum energy of 
photoelectrom on frequency. 



are ejected from a solid (or liquid) surface into a surrounding 
vacuum. Photomultiplier tubes use this effect. 

Electrons and ions may he produced in a gas by ptiotoionista- 
tion. The ionization chambers used to detect x rays utilize this 
effect. 

Conduction electrons and positive "holes" which remain 
inside a solid may be responsible for either photoconduction or 
a photovoltaic effect. Photoconduction is a decrease in resistivity 
under the influence of radiation. It is used in television camera 
tubes and in control devices where an external battery furnishes 
the electric power. The photovoltaic cell is a device for converting 
radiation into electrical power. Kadiation acting on two dissimilar 
layers in the cell gives rise to an emf in much the same way a 
voltage is produced when Cu and Zn plates are dipped in acid in a 






Electromagnetic Radiation 175 

voltaic cell. Photovoltaic cells are used in photographic exposure 
meters and in solar batteries. 

There is also an internal photoelectric process, within an 
atom, called the Auger effect, or autoionization. An x-ray quan- 
tum may be absorbed within the same atom from which it origi- 
nates, with the ejection of one of its electrons. The net effect is 
thai the atom adjusts from an excited level to a lower-energy 
level, with the emission of an electron. Finally, there is an inverse 
photoelectric effect in which an electron is absorbed by a solid 
and a photon emerges. 

The photoelectric effect gives strong support to Planck's 
hypothesis that light of frequency v can be emitted or absorbed 
only in packets of energy hv. The citation which accompanied the 
award of the Xobel Prize to Einstein stated that it was for "his 
attainments in mathematical physics and especially for his dis- 
covery of the law of the photoelectric effect." 



13.10 THE CONTINUOUS X RAY SPECTRUM 

X rays are electromagnetic waves of very short wavelength, 
about 10"" to I0~" m. In an x-ray tube (Fig. 13.1")) a battery li 




Fig, 13.15 An x-roy tube. 

heats a tungsten filament C so that it emits electrons. A potential 
difference of several thousand volts between cathode (' and target 
T accelerates the electrons. The fast-moving electrons are quickly 
decelerated when they strike the metal target. Most of their 
energy is converted into heat by collisions with atoms of the 
target. But as the electrons are decelerated, they are expected to 



176 Looking In: Atomic and Nuclear Physics 

radiate, according to Maxwell's electromagnetic theory. The 
Gorman term Itrrmxxtniltliniu is used for this "slowing-down 
radiation." The radiation is emitted in all directions. When one 
examines the beam of x rays emerging from the hole in a lead 
shield, one finds a continuous distribution of frequencies up to a 
certain maximum. This maximum frequency depends on the 
potential difference at which the tube is operated: v,„„ x /AV = a 
constant for a wide range of voltages. 

The high-frequency limit in the continuous x-ray spectrum 
is difficult to explain classically. It is easily clarified by the photon 
hypothesis. An electron may suffer numerous decelerations as it 
encounters various atoms in the target. ICach time a photon is 
produced, whose energy hi> is equal to the decrease in kinetic en- 
ergy A A'* of the electron. Clearly, the highest-frequency photon 
that can be produced is that which results from the complete. 
conversion of the electron's kinetic energy into a single photon. 
Since electrons arrive at the target with energy e Al". 



hv mnx = e AV 



(13.35) 



From this Duane and Hunt law, Eq. (13.35), r'h may be deter- 
mined from the sharp cutoff of the x-ray intensity versus fre- 
quency curve at p„ k , x . There is good agreement with the ratio e/h 
determined in other ways. 

13.11 THE COMPTON EFFECT 

Another even more direct confirmation of the photon hypothesis 
came about |<)23 in A. H. Compton's explanation of properties of 
scattered x rays. Compton allowed a beam of monochromatic 
x rays to fall on a hlock of scattering material such as carbon. 
The scattered radiation was examined in an x-ray spectrometer 
(an instrument which uses » crystal and an ionization chamber 
to measure the wavelength of x rays incident on it J. 






Spectrometer /-\ 
C X 



Fig. 13,16 Apparatus for observing 
Compton scattering of x roys. 






Electromagnetic Radiation 177 

According to classical theory, scattered radiation should have 
the same frequency as the incident radiation. Compton found 
such unmodified radiation, but in addition he found a scattered 
wavelength X' greater than that of the incident beam. The shift 
In wavelength X' - X was found to increase as the angle 8 at 
which the scattering was observed was increased (Fig. 13.17). 
The scattering of x rays with increase in wavelength is called the 



Fig. 13.17 Wavelength shift in 
Compton scattering of x rays. 




Wavelength, X — *" 



178 Looking In: Atomic and Nuclear Physics 

Campion, effect. The. plioton hypothesis provides a straightforward 
explanation. 

We describe the Compton scattering as an elastic collision of a 
photon with a free electron which is at rest before the collision 
(Fig. 13.18). We ascribe to the photon the "equivalent mass" 
hvfe* (Chap. 14), and to this mass we attribute linear momentum 
hv/c. The conservation of momentum may be stated in two 



Photon 



* Before After 

Fig. 13.18 Compton scattering of a photon. 




Electron 



equations, since momentum is a vector quantity and the law of 
conservation applies to each of the components: 



kv hv „ , 
X component: — = — cos 8 + me cos # 
c c 

hv' . 
y component: = — sin $ — mv sin 4> 



(13.30) 
(13.37) 



where v' is the frequency of the scattered photon and me is the 
momentum of the recoil electron. 

Conservation of energy requires that 



hv = hv' 4- A* 



(13.38) 



where K k is the final kinetic energy of the electron. Solution of 
these three equations provides an expression for the wavelength 
shift 



X' - X = -- (1 - cos 9) 



mr 



(13.39) 



which agrees with experimental data. The unmodified radiation 
is interpreted as due to photons scattered by electrons strongly 
bound in atoms. 



Electromagnetic Radiation 179 

Example. What is the change in wavelength <>r x rays Com|ii on-scattered 
in the backward direction (6 = 180°)? 



X' -X = 



6.625 X 10 3t joule sec 



9.1 X 10 3l kg (3.0 X IIP in sci") 
= 0.048 X IO" 1 * m = 0.0484 A 



(1 - cos 180°) 



13.12 WAVE-PARTICLE DUALITY: PROBABILITY 

We have discussed two theories of electromagnetic radiation. 
The classical theory says iliat radiant energy flows continuously 
as a wave. The wave theory gives a satisfying explanation of 
interference, diffract ion. and polnrixa! ion experiments. The quan- 
tum theory says that radiant energy is exchanged in quanta of 
amount Ac, whose value depends on the frequency c of the light. 
This photon theory gives a satisfactory interpretation of many 
experiments in atomic physics (blackbody radiation, photo- 
electric effect, the frequency limit in a continuous x-ray spectrum, 
the Compton effect, and the line spectra characteristic of 
elements). 

In some ways these two theories are mutually contradictory. 
The wave theory says that the photoelectric effect should show a 
time lag when the light source has a very low intensity. The 
photon theory when used to explain a single-slit diffraction pat- 
tern would have to assert that these particles arriving at certain 
points on the screen would "cancel" each other. (How?) Although 
each theory works well for its own experiments, something has 
to "give" when we try to put the two theories together. 

A resolution of this tlillieulty was suggested by a novel idea 
proposed by Louis de Broglie, in his 1'h.D. thesis in 1924. From 
consideration of relativity theory (Chap. 14), he deduced that all 
particles must have a wave nature, just as light has a wave 
nature. The intensity of the particle wave at any given point (or 
the square of the wave amplitude) is interpreted as proportional 
to the probability of finding the particle at that point. The 
de Broglie relationship for the wavelength X of a matter wave is 



X = * 

V 



(13.40) 



where p is the momentum (mv) of the particle and h is Planck's 
constant. 



rr 1 * 



180 Looking In: Atomic and Nuclear Physics 

The exploration of wave-particle duality was continued in 
M)28 by Max Horn. as follows. Kuergy is not distributed continu- 
ously throughout an electromagnetic wave: the energy is carried 
by the photons. The intensity of the wave (which the classical 
theory defines as energy flow) at a point in space is really a meas- 
ure of the probability of finding a photon there. The classical wave 
has become a sort of guide for the individual quanta of energy. 

We have resolved the wave-particle dilemma, but at the cost 
of admitting that laws of chance govern the motion of micro- 




CDBEAEBDC 
Fig. 13.19 Diffraction at a single slit. 




Fig. 13.20 Double-slit diffraction. 






scopic particles. If we photograph a single-slit diffraction pattern 
(Fig. 13,19), the relative intensities tell us that many photons 
struck the plate in the region .4, a fair number in li, some in C t 
etc. Very few hit near I) or /■.'. Suppose, now, we perforin the 
experiment with one photon. We can predict that it has a high 
probability of hitting near .1. a fair chance for li. less for ('. very 

little chance of hitting near i) or K, Hut we 08JQ prediel to 

which point the photon will actually go. 

Consider an experiment in which a beam of electrons falls on 
two slits (Fig. 13.20). The electron distribution at I 1 predicted 



Electromagnetic Radiation 181 

by classical theory is shown by the dashed line. The distribution 
actually observed (solid line) is that predicted by considering 
interference of the tie Hroglie waves. Now consider one electron 
shot at a time. According to this wave picture, each electron is 
represented by a single wave packet which divides equally be- 
tween the two slits. Vet if we place a particle detector at slii .1, 
we never observe half an electron: we find cither a whole particle 
or no particle. It is intriguing to try to devise an experiment that 
would reveal the slit used by individual electrons, without de- 
stroying the interference pattern. No one has succeeded. If a 
detector is placed at A, the interference pattern smooths out; the 
classical result is obtained. 



13.13 THE UNCERTAINTY PRINCIPLE 

A consequence of quantum theory is that one cannot determine 
simultaneously the exact position and velocity (or momentum) 



Fig. 13.21 Supermicroscope. 




o/vw 



of any particle. As an example, assume that the exact momentum 
of a particle is known. Then it has a definite wavelength X = h/p 
and is a continuous plane wave of uniform "intensity." It is equally 
probable to find the particle anywhere in space. At the other 
extreme, assume that we have located the particle within a very 
small region of space. Then its wave function is a short packet 
that does not have any unique X. Hence the momentum is fuzzy. 
The uncertainty principle predicts that in general we cannot 
make a measurement on a system without disturbing it. for 
example, suppose we try to "view" an electron with an (imagi- 
nary) supermicroscope .1/ (Fig. 13.21) to determine its position x 
and momentum p. We may borrow an expression from optics 
which says that the smallest displacement A.v the instrument can 



182 Looking In: Atomic and Nuclear Physics 

defect depends on the wavelength of the light and the half-angle 
a subtended by the objective lens: Ax = X/(a sin a). We "view" 
the elect ion by light which (-liters the microscope anywhere within 
angle 'la. This radiation, scattered by the electron, makes a con- 
tribution to the electron's momentum which is unknown by 



Apz = p sin a — ^ sin a 



If we write Ax as the uncertainly in position of the electron and 
Ap* as the uncertainty in its momentum, combining the last two 
equations gives 









1 / X \ (k . \ h 

Ax Ap x = I -. — 1 [ :- sin « J = - 
a \sm a/ \\ / a 



(13.41] 



which shows that as we increase the precision of our measurement 
of 3% the value of p becomes subject to greater uncertainty. (The 
foregoing is offered merely to amplify the statement that wc can- 
not make a measurement on a system without disturbing it. The 
numerical value of a depends on the criterion used for resolving 
power.) 

Werner Ileisenberg formulated the uncertainty principle in 
1927 showing that from Sehrodinger's equation (Chap. Hi) /i/4ir 
is the lower limit of the product of simultaneously measured 
value of a particle's position and momentum: 



Ax Ap > -r- 



(\:i.m 



Mere Ar and A/; are defined as vm- deviations. There is an uncer- 
tainty relation only between certain pairs of variables, those 
which are "canonically conjugate variables."* There are uncer- 
tainty relations, for instance, between position and momentum 
(discussed above), angular momentum and angle, and energy 
and time. 

We have seen that quantum theory is significantly different 
from classical theory in dealing with the interactions and struc- 

* See Condon and Odtshaw (wis.), "Handbook of Physics," cinq). I>, 
McGraw-Hill Book Company, Inc., New York, 19oX. 



Electromagnetic Radiation 183 

tore of small particles, IT the quantum theory is correct, as we 
think, there is no hope of understanding the elementary structure 
of matter (atomic and nuclear physics) from the viewpoint of 
classical physics. In the following chapters we shall use the ideas 
of quantum theory. It will be interesting to see, however, that 
there is a region between macroscopic and microscopic physics 
where the laws of classical and quantum physics smoothly overlap 
(correspondence principle). 



The secret of education lies in respecting the pupil. R. W. Emerson 



The most essential characteristic of scientific technic is that it proceeds from 
experiment, not from tradition. Bertrand Russell 



The most brilliant discoveries in theoretical physics are not discoveries of 
new laws, but of terms in which the law can be discovered. 

Michael Roberts and E. R, Thomas 






Relativity Wonderland 185 



14 



tivistic mechanics. The insight it gives into the binding energy of 
nuclei and the liberation of nuclear energy will he our cbier 
interest. We shall look, also, at what relativity says about simul- 
taneity of events, time dilatation, and the aging of voyagers in 
spaceships. 



Relativity 
Wonderland 



The supreme task of the physicist is 
to arrive at those universal elemen- 
tary laws from which the cosmos can 
be built up by pure deduction. There is 
no logical path to these laws; only 
intuition, resting on sympathetic 
understanding of experience, can 
reach them. A. Einstein 






14.1 NEWTONIAN RELATIVITY 

In elementary experiments in mechanics we recognize that trans- 
lator? motion can be measured only as motion relative to other 
material bodies, such as the workbench or the earth. Measure- 
ment of a speed involves measurement of both distance and time. 



vt 



S' 






Fig. 1 4.1 Reference system S' moves with constant velocity v 
in x direction relative to reference system S. 






Relativity and quantum mechanics are two great theories of 
twentieth-century physics which have modified in remarkable 
ways our ideas of the physical universe. For bodies traveling at 
speeds close to the speed of light, Newtonian mechanics is replaced 
by rclativistic mechanics. The relativity theory of the physical 
meaning of space and time makes some simple predictions of 
great importance. (I) The mass of a particle is shown to be vari- 
able, depending on the speed of the particle; (2) it is impossible 
for any particle to have a speed greater than the speed of lighl ; 
(3) mass and energy are interconvertible. In this chapter we shall 
consider the evidence which leads to the formulation of rela- 

184 



Up to now we have perhaps intuitively regarded time as a unique 
variable, quite distinct from, say, space, energy, or the* behavior 
of material things. \Ye might agree with Newton that "Absolute, 
true, and mathematical time, of itself, and by its own nature. 
Hows uniformly on, without regard to anything external." It 
is helpful to formulate the kind of relativity implied by these 
ideas for later comparison with the new relativity. 

Consider a material reference body and some sort of timing 
device (the rotating earth, or a crystal oscillator) to constitute 
a space-time system of reference for making measurements to 
locate particles or to describe events. Now suppose a second 
system of reference S' (Fig. 14.1) to be in uniform motion with 
respect to the first reference system >S\ along the common line of 
i heir x axes. Let the velocity of S' relative to S be v. Let us agree 
to reckon time from the instant at which the two origins of coord i- 



186 Looking In: Atomic and Nuclear Physics 

nates and 0' momentarily coincide. At any later time, the co- 
ordinates of 0' measured in system S will be x = vt, y = 0, 
z = 0. An event which occurs at coordinates x,i/,z and time / 
in system A' will, according to Xcwlon, have coordinates in system 
S' given by 



v* - y 



X = z 

f = t 



(14.1) 



14.2 THE AETHER 

The wave properties of light were, demonstrated by Young, 
Fresnel, and others during the first part of the nineteenth century 
and were explained in Maxwell's brilliant theory of electromag- 
netic radiation (Chap. 13), It was difficult for scientists of the 
nineteenth century, as for us, to conceive of a wave motion 
without a material medium to transmit its vibrations. So they 
invented a medium called the aether for the propagation of light. 
The aether was thought to pervade all space, as well as trans- 
parent material bodies. The assumed existence of the aether sug- 
gested two interesting consequences worthy of experimental 
check, (I) Light waves should travel with a definite speed 
(c = 3.0 X 10" m see in "empty" space) with respect to the 
aether itself. Then the apparent speed of light relative to a mate- 
rial body moving through the aether should be different from c 
and should depend on the speed of the body. (2) An "absolute" 
velocity of the earth or any other body should be ascertainable 
from measurements on tight waves transmitted through the 
aether. 



14.3 MICHELSON'S INTERFEROMETER 

An experiment designed to detect the motion of the earth relative 
to the aether would require very sensitive apparatus, for the orbi- 
tal speed of I he earth is only about III ' the speed of the light 
signals that would be used in the measurement. With this prob- 
lem in mind, Miehelson devised an interferometer, an instrument 
in which interference patterns produced by two light beams are 
used to reveal differences in the optical paths of the beams. 
When the optical paths (Fig. 14.2) happen to he equal, beams 1 



Relativity Wonderland 187 

and 2 will arrive at E in phase and produce a bright field, by con- 
suuetive interference. As distance .1.1/ is increased X I by moving 
mirror .1/, the optical path for ray 1 is lengthened by X/2, and 
destructive interference of rays 1 and 2 at li gives a dark field. 



r 



c\ 



3-w 



~G\ 



M 




Source 
Fig, 14,2 Michelson's i nf erf ero meter. 



Mirrors .1/ and .1/' are set nearly but not quite perpendicular to 
produce a field crossed by alternate bright and dark interference 
fringes (Fig. 14.8). These are counted as they move past a refer- 
ence mark R, For each fringe that passes the mark, the optical 
path has changed by one wavelength. This change might he pro- 
duced by moving .1/ a half wavelength. However, the change 
might also he produced by a change in the speed of light in beam 
I (on the substitution, for example, of a gas of different index of 
refraction for the air in that one beam). 



Fig. 14.3 Fringes and a reference mark. 



188 Looking In: Atomic and Nuclear Physics 



14.4 THE MICHELSON -MORLEY EXPERIMENT 

Assume that the earth travels through stationary aether with a 
speed i> and that light has a speed c in the aether. Consider a 
Mieheison interferometer arranged so that one of its two equal 
arms is parallel to the earth's velocity {Fig. 14.4). Then the. times 
required for the light beams to travel the distances AM A and 
AM' A will be unequal. The speed of a beam traveling from .1 to 
.1/ is c — t" relative to the interferometer. On the return from .1/ 

M' 



i 

A 



4—W£ 



i 



M 




n 

HI 

u 



Fig. 14.4 Light poths in moving interferometer and velocity 
vector diagram. 

to A, the speed of the beam relative to the interferometer is 
c + v. The time for the round trip AM A is thus 



c — V C + V 



2w 



(14.2) 



Since r is small compared with c, we may use the binomial 
theorem to obtain the approximation 



1 — rye 2 c \ c* / 



(1-U) 



A wave front leaving A toward mirror .1/' will be returned, 
according to Huygen's principle, but only after A has moved to 
a new position .]'. The component of the velocity of light in the 
direction perpendicular to the motion of the interferometer is 
VV 1 — v i . The time for the round trip AM' A is 



k « 



2s 



2»/ 



vV - v* Vi 



!n /; _ 2* / 



Iff* 

2 ri 



1+3^ + 



) 



(14,1) 



Relativity Wonderland 189 

Waves which are in phase when they reach A from the mono- 
chromatic source will differ in phase when they return to .4 after 
reflection, because of the time difference: 



Ai = ft — h = 



si- 



(14. o 



If the interferometer is rotated 90°, paths t and 2 will have 



their roles interchanged and the total retardation will be 2sp ! 
The number of fringes passing the reference mark should be 



N = 



path difference _cA/ = c2st>* _ 2siP 
wavelength X Xc* c 2 X 



(14.6) 



To estimate the magnitude of fringe shift to be expected, we 
may assume that the earth's velocity through the aether is the 
same as its orbital velocity, about HO km sec. By using multiple 
reflections, .Mieheison and Morley attained an effective path s of 
10 m (Fig. 14.5). For light of wavelength 5,000 A we should then 
estimate a maximum fringe shift of 

_ 2 X 10m(3 X lOWsec)' _ f . ( 4 ?) 

N ~ OTX 10«m/»»)»(5.0 X 10- T m) U * lr " lge U4 ' J 

A fringe shift of this amount is readily detectable with the 
apparatus. It should then be possible to measure the fringe shift 
and from it compute the velocity of the earth relative to the 
aether, that is, the absolute velocity of the earth. 

Surprisingly. Mieheison and Morley found no fringe shift 
when the interferometer was rotated in a pool of mercury. It- 
appeared that optical experiments cannot detect motion of the 
earth relative to the aether. 

Mieheison and Morley reported their results in 1887. No 
subsequent experimental evidence contradicts them. Some linger- 
ing doubts were laid to rest in a review article published in the 
Reviews of Modern Physics (pages 107- 178) in 1955. 

Several attempted explanations for the apparent impossibility 
of measuring the earth's absolute motion failed to gain acceptance 
when they either did violence to established theory, disagreed 
with known astronomical data, or introduced too many special 
hypotheses. 










m 











Fig. 14,5 Michelson interferometer designed to detect "absolute motion" of 
the earth, (o) Interferometer was mounted on a stone, ond floated in mercury 
to damp vibrations and to permit rotation, (b) Multiple reflection of beams gave 
an effective path length of 10 m. 






190 



Relativity Wonderland 191 

14.5 POSTULATES OF THE SPECIAL THEORY 
OF RELATIVITY 

Consider several physicists in a completely enclosed elevator or 
railroad car, moving with constant velocity relative to earth. 
Could these people detect and measure the velocity of their 
enclosure from observations made inside, with pendulum, spring 
balance, etc? 

Proceeding from considerations such as these, Henri Poincard 
in the period 1 SiM) to 1904 developed the hypothesis that it is 
impossible to determine absolute motions of a body or of a refer- 
ence system by any dynamical, electromagnetic, or optical means. 

Measurement of the velocity of bodies relative to a stationary 
net Iter seemed to Iks the best device classical physics could offer 
for determination of "absolute" motion. The negative result of 
the Michelson-Morley experiment was interpreted by Einstein 
as indicating that only relative velocities can be measured. Con- 
sequently, the general laws of physics must be independent of the 
velocity of the particular reference system of coordinates used to 
state them, otherwise it would be possible to ascribe some abso- 
lute meaning to different velocities. 

The special or restricted relativity theory of 1905 was limited 
to consideration of reference systems moving at a constant 
velocity with respect to each other {Fig. 14.1). Einstein based his 
theory on two postulates: 

1. The laws of physical phenomena are the same when stated in 
terms of either of two reference systems moving at constant 
velocity relative to each other {and can involve no reference 
to motion through an aether). 

2. The velocity of light in free space is the same for all observers 
and is independent of the velocity of the light source relative 
to the observer, 

{The "general" theory of relativity, 1 9 Hi, is Einstein's theory 
of gravitation and will not be considered here.) 

Suppose person A. at rest in a laboratory, assigns to every 
event which he observes a position (j-.i/a) relative to a particular 
origin fixed in his laboratory and a time J as indicated by a clock 
at rest in his laboratory. Now let person B move through A's 
laboratory with speed u in A's positive x direction. Let person 



192 Looking In: Atomic and Nuclear Physics 

B measure positions relative to an origin moving with him and 
times with a clock (just like A's clock) also moving with him. 
Then to each event B will assign a position (x',y',z') and a time I'. 
Assume that the clocks are synchronized to read I = (' = when 
the (x'y'z') axis momentarily coincides with the (xi/z) axis. The 
relations which connect the distance and time intervals between 
two events as measured from the two inertial reference frames are 



x = — 



V = y' 
z = z' 



x' + lit' 



== X' = 






U' - !J 

Z' = 2 



X — ut 



» _ t_— (u/c*)x 

Vi - 1* 1 /** 



(14.8) 



These transformation equations were developed by Voigt 
(1887) and Lorentz (1904) in exploring the aether hypothesis. 
But Einstein showed that the transformations satisfied his rela- 
tivity hypothesis that the speed of light will be the same in each 
coordinate system. 

Example. Show that light has speed c in both the 8 and S' coordinate 
systems. 

Suppose that the light starts from x = 0, y = 0, t = at J = and 
moves in the positive x direction. It will arrive ill the point x = A' at 
the time X/e, time its speed through the laboratory is ,-.. Person B will 
observe the light to arrive at the point 

x , m X - u(X/c) 

Vl - u7c* 

at the time 

t , m X/e - (uM )X 

Vl - m*/c* 

The speed of light in the S' coordinate system is thus 

x' X - (u/c)X 



V = -7 = 



I' X/e - (u/c*)X 



= r 



Relativity Wonderland 193 

transport energy from one point to another with a speed exceeding 
the speed of li<iht. 

Several relations of particular interest will now be discussed to 
illustrate the meaning of space and time variables. 

14.6 VELOCITIES NEVER ADD TO MORE THAN c 

Suppose that our two observers in coordinate systems S and S' 
both observe an object which Hies past in the x direct ion. Observer 
B measures the speed of the object relative to him as v' = dx'/dt'. 
If we express v' in terms of the coordinates of the laboratory 
observer A, we find 



dt' 



d[(x - mQ / V'1 - u */c*] dx -udl 

d[[t - (*/c*)*]/Vl - »Vc*l di ~ (*/<&* 

-r^ws (l4! » 

where in the last step numerator and denominator were divided 
hy dt and dx/dt = » was written for the speed of the object in the 
laboratory. Thus we have 

»' + « 



V = 



V — u 



or 



v = 



1 + uv'/c* 



(14.10) 



1 — uv/c 1 

The speed v relative to the laboratory is not, as we might have 
expected, exactly equal to the speed v' relative to B plus the speed 
u of B relative to the laboratory. 

Example. While observer B is moving through the laboratory with 
speed u = 0.90c, a flying object passes him with a speed which he meas- 
ures as v' = 0.90c. What is the speed of the flying object relative to the 
laboratory? 



v = 



V ' + « 



0.90r + 0.90c 



I.SOc 



1 + w'/c* 1 + (0.90c) (0.90c)/c ! 1 + 0.81 



= 0.994c 



In other words, if a car were traveling at speed 0.90c, you would have 
to drive at a speed of only 0.994c to pass it with a relative speed of 0.90r! 






In a mathematical sense, the principle of relativity is that the 
equations of physical phenomena must be invariant in form under 
Lorentz transformations. The basic physical assumption of rela- 
tivity is that no mechanical or electromagnetic influence can 



14.7 WHAT DOES "SIMULTANEOUS" MEAN? 

Einstein pointed out by the following railroad story that man 
cannot assume that his sense of "now" applies to all parts of the 



194 Looking In: Atomic and Nuclear Physics 

universe. He pictured a straight section of track with an observe? 
seated on an embankment beside it. During a thunderstorm, two 
lightning bolts strike the track simultaneously, at points Xi and 
.)■•:. Kiustein asks: What do we mean by "simultaneously"? 




x i A 

Fig. 14.6 My time is not necessarily your time. 



Assume that the observer is seated midway between r, and .<-._>, 
Assume that he has arranged mirrors so he can see x t and z» at 
the same time without moving his eyes. Then If the reflections of 
the lightning flashes are seen in the mirrors at precisely the same 
instant, the flashes may be regarded as simultaneous, by ob- 
server A. 

Now assume that a train speeds along the track and that 
observer B on the train sits in an observation dome, with an 
arrangement of mirrors for viewing points x x and x->. It happens 
that observer B finds himself directly opposite A when lightning 
strikes the rails at a and x-,. Will the flashes appear simultaneous 
to B? No, for if his train is moving from x-, toward .c,, then the 
flash at .10 will be reflected in his mirrors a fraction of a second 
later than the flash in .r,. (In the limiting case with a train travel- 
ing at speed e, B would never see light from x«.) Whatever the 
speed of the train, the observer B on it will always say that the 
lightning Hash ahead of him has struck the track first. 

In generalizing, we are forced to admit that two events which 
occur at different places may be simultaneous for one observer 
and not simultaneous for another. We cannot assume that a 
single time scale (( = (') can be used with any and all coordinate 
systems. 



14.8 THE FITZGERALD-LORENTZ CONTRACTION 

To explain the null result of the Michelson-Morley experiment, 
Fitzgerald in 1893 arbitrarily assumed a contraction of the arm of 
the interferometer in the direction of motion of the apparatus. 



Relativity Wonderland 195 

The special theory of relativity predicts the same contraction 
but ascribes it to the relative motion of the body and the observer. 
Consider a material object in coordinate system S' whose 
surface may be defined by the relation 4>{x' ,tf ,z') = 0. Then, by 
the Lorentz transformation, the form of the. surface as viewed in 
coordinate system S is 






In particular, suppose that a spherical surface of radius a is 
described in system S' by (Y) 1 + Cv') 2 + {*')" ' a" = <>■ Tbfe 
appears in system S to be a moving ellipsoid 



( x r*J + K 



+ 3-1 



whose semiaxes are (a \/\ - u- c-,a.a). The surface undergoes 
contraction in the direction of motion in the ratio y/\ — u s /c 2 : 1. 



14.9 TIME DILATATION: THE CLOCK PARADOX 

Consider now ihreflVel of relative motion on a flock. Two events 
occur at a point in coordinate system .S": one at time t\, the other 
at a later time 4 T <> an observer in S these events take place at 
different points in space, (.ri,//,s) and Oj.(/,z), as well as at differ- 
ent times, such that {x» - xi) = u(t t - h). Prom the Lorentz 
transformations 



tt - h = 






Thus the sequence in time of the two events is the same, but &t 
appears longer for the observer in S than for the observer in .S". 
This is interpreted as meaning that a moving clock appears to 
run at a slower rate than does an identical clock at rest, in the 
ratio Vl - «7c°-:l. 

The imminence of space travel has revived interest in the 
"clock paradox" or "twin paradox." One of two identical twins 
leaves his brother on earth and voyages at high speed into dis- 
tant space. On his return, he finds that his brother has grown 






196 Looking In: Atomic and Nuclear Physics 

much older than he, because of time dilatation in the spaceship. 
Superficially, this is a paradox, for it challenges "common sense." 
Also, it seems to contradict the assertion of special relativity that 
in describing physical events all observers are equivalent; none 
has a preferred or absolute reference system. The aging or clock 
effect seems to provide a way of distinguishing among observers. 
But, relativity asserts the equivalence of observers in inertial 
systems, and since one of the twins accelerated at the start of his 
space trip and again when he altered course to return, he did not 
view his brother from the same inertial system before and after 
the trip. So there is no paradox. 

The intriguing question remains: Did the stay-at-home 
brother grow older faster? Yes. In his 190") paper "On the Elec- 
trodynamics of Moving Bodies," Einstein wrote, 

If at the points A and B there arc I ho stationary clocks which, 
viewed by a stationary observer, arc synchronous, and if the clock at A 
is moved with the velocity v along the line AB to B, then on its arrival 
at B the (.wo clocks no longer synchronize, but the clock moved from 
A to B lags behind the other which has remained at B by tv 3 /2c- (up to 
magnitudes of fourth and higher orders), ( being the time required for 
the journey from A to B. 

It is at once apparent that this result still holds if the clock moves 
from A to B in any polygonal line, and also when the points A and B 
coincide. 

Bergman n has suggested the following elucidation of the clock 
effect, tig. I 1.7. 



9"-^= — I" 



Fig. 14,7 Clock paradox. 

Observer A arranges for periodic light signals to go from 
lamp L to mirror .1/ and back (a kind of optical clock). Light 
travels a distance 2/) for each LM L circuit. Observer B is moving 
with constant speed u at right angles to the line LM. For him, 




Relativity Wonderland 197 



the same light signal travels the larger distance 2D'. If observer 
B set up a similar experiment in his coordinate system S', his 
light signals would complete their round trips in shorter times 
than noted by observer A. The discrepancies arise because the 
two observers do not agree on which of two distant events {com- 
pletion of the nth round trip by either light signal) takes place 
first. 

Now let observer B suddenly reverse his velocity (u[). He 
is now in a different Lorentz frame. (He accelerated.) His notions 
of simultaneity have changed. Observer A sees B coming toward 
him, with B's light signals arriving slower than his own. When 
they meet, A's signals have completed a larger number of LM L 
circuits than have B's signals. Observer A has aged more than B. 






Example. What, will he the difference in I he rates of two identical 
clocks, one of which is on a spaceship moving at 300 mi sec relative u> 
the other? 



u = 300 mi/sec = 5.25 X 10 s - m/scc c = 3.0 X 10 a m/sec 
Relative change in rate 

f. u* f, S^iTx 10" / - — - — 

= 0.002 per cent, approx. 



10"« 



Experimental detection of time dilatation was achieved by 
Ives and Stilwell (1938) on viewing the spectral lines of hydrogen 
atoms which were given a high speed directed away from the spec- 
troscope. An arrangement was used to distinguish relativity 
effects from Doppler effects. Light from the atoms fell on the 
spectrograph slit directly, and also after reflection in a mirror set 
at some distance and normal to the velocity of the atoms. Owing 
to the Doppler effect, each spectrum line was split into two fre- 
quencies. Then light from hydrogen atoms at rest was viewed 
with the same spectrograph. This gave lines slightly displaced, 
in frequency, from the middle, of the Doppler pairs, in amount 
predicted by relativity, 

.Measurements of the lifetimes of mesons have been used to 
cheek relativity predictions. The mean life of fi mesons (about 






198 Looking In: Atomic and Nuclear Physics 

'2 X 10"* sec) has been found to depend on I heir speed roughly 
in (lie way predicted by relativity. 



14.10 MASS AND ENERGY 

Two results of relativity theory which are of especial importance 
in atomic physics are (1) tlie variation of the mass of a particle 
with its speed and (2) the equivalence of mass and energy. 

Experiments have been performed, first by Bucherer in 1909, 
on the deflection by a magnetic field of electrons whose speeds 
are not small compared with the speed of light. The acceleration 
may be determined from the radius of curvature of the path 
a = v*/R. The force producing this acceleration is the magnetic 
side thrust on the electron / = Bev. It is found that for high- 
speed electrons Newton's law in the form/ = ma is not satisfied. 
But Newton's law written in the form / = d(mv)/dt is satisfied, 
provided we assume that the mass m of the particle depends on 
its speed. It is found necessary to assume that a particle which 
has mass m u when at rest has a mass 



m = 



m-o 



y/\ - i.'Ve* 



(14.11) 



when moving with speed v. The quantity wio is called the rest 
mass. When v « c, m = m-u. 

Variation of mass with speed is accepted in relativity theory 
as requisite for the conservation of momentum, which remains a 
basic principle of mechanics. In order to have the total momen- 
tum of an isolated system remain constant, the momentum of a 
panicle is delined as 



p = mv = 



Met) 



Vl - »Ve* 



(14.12) 



Table 1 4. 1 shows for various ratios of v/c the kinetic energy of 
an electron, the ratio vi/m , and the product BR, from which one 
may get the radius R of the path in a magnetic induction B of 
given value. Looking at the table, one might say that in problem 
solving to slide-rule accuracy, one can neglect relativity variation 
of mass for bodies having speeds less than 0. 1 l he speed of light. 






Relativity Wonderland 199 



Table 14.1 Date on electrons 



V c 


Energy, ev 


ffl fllfl 


8R, X10 • weber/m 


0. 





1.00000 





0,0100 


25.54 


1.00005 


17.0 


0.0200 


102.2 


1.00020 


34.06 


0.0500 


638.5 


1.00125 


85.0 


0.100 


2,575 


1 .00504 


171.3 


0.200 


10,530 


1 .02062 


347.8 


0.500 


79,030 


1.1547 


983.6 


0.600 


127,700 


1.25000 


1,278 


0.700 


204,300 


1.4002 


1,669 


0.800 


340,500 


1.6666 


2,272 


0.900 


661,000 


2.2941 


3,517 


0.990 


3,110,000 


7.0888 


11,960 



The kinetic energy of an object having speed r is equal to the 
energy required to accelerate it from rest to the final speed v. 

E h - //-'(cos $) ds = jT dx (14.13) 

But now we must use in place of/ — ma for Xewtou's second law, 

d me 



F- 



givmg 



dt y/l - B */ c » 



ft- [* »g_ d X = m f r vd( ^=) 

J dt y/\ - r * J3 JO VVl - W/ 

This may be integrated by parts using the standard form 

/« dv = uv — $v du 
to obtain 



ft = ffloiM — 7= - 1 J 



ft = »ic 2 — muc 2 



(14.14) 

(14.ir>) 

This expression replaces the classical formula hn^ for kinetic 
energy when v is comparable with c. 

The equation for kinetic energy, Kk = (m — m.n}c' i , says that 
when we speed up a particle, the increase in energy is propor- 
tional to the increase in mass of the particle: 

&S = c-(Am) (14.10) 



200 Looking In: Atomic and Nuclear Physics 

We can identify c- times the relativist ie mass of the particle with 
the total energy K of the particle: 

E = mc * = A* + Wo c 2 (14.17) 

Total energy = kinetic energy + rest energy 

Kinstc.in's famous relation IC = mc % states that mass and 

energy are different aspects of the same thing. It tells us tiie rate 

at which one may he converted into the other. 

Example. Find the energy equivalent of 1 gm of coal (or any other 
substance). 

E = me 2 = 0.001 kg (3.0 X 10* m/sec,} 1 = 9 X 10 13 joules 

= 25,000 niogawutt-lir 

Only a liny fraction of this amount of energy is released in the burning 
of I Km of coal: tin- combustion products have a mass only sJightly less 
than 1 Kin. In nuclear reactors, a somewhat larger percentage conversion 
takes place, but it is still a small fraction. 

14.11 NUCLEAR BINDING ENERGY 

Mass spectrograph measurements show that the mass of any 
stable isotope is less than the sum of the masses of its constituent 
protons, neutrons, and electrons. Kinsfein's mass-energy relation 
suggests thai the mass discrepancy might account for the energy 
needed to hold a nucleus together, against the dispersive forces 
exerted by the protons on each other owing to their positive 
charges. 

The mass of the constituent particles for nucleus zX. A is the 
sum of Z proton masses and (A-Z) neutron masses. The mass 
defect Ant is then calculated from 



Am = Zm.fi + (A — Z)m„ — M z,A 



(14.18) 



where ma = 1.00814o amu, mass of the hydrogen atom 
m„ = 1.00898b' amu, mass of the neutron 
M x , = mass of the neutral atom of atomic number Z and 
and atomic mass number .1 
From A/i = (Awi)c 3 one can calculate that I amu is equivalent 
to Oil I Mev (million electron volts). 

Binding energy is primarily a property of the nucleus. Yet in 
the equation above we have used data for neutral atoms >»u and 






(14.19) 



Relativity Wonderland 201 

Mx..\ which of course include electrons. We justify this procedure 
by the following facts: (1) If a nuclear reaction is written in terms 
of the symbols for the corresponding atoms, the number of elec- 
trons on one side of the equation generally cancels the number of 
electrons on the other side. (2) The minute changes in mass 
which may accompany the formation of an atom from its ion and 
electron(s) is negligible. (3) The mass data from mass spectro- 
graph experiments are always tabulated in terms of neutral atoms 
(e.g., Na) even though deflection measurements must be made on 
ions (e.g., Na ++ ). It is to avoid the trouble of specifying each time 
the degree of ionization that the experimenter adds to his experi- 
mental value for the mass of the ion the proper number of electron 
masses and reports as the isotope mass the* computed mass of the 
neutral atom. 

The binding energy per nucleoli is defined as the binding 
energy divided by the number of nuclear particles: 

Binding energy _ Am <■- 
Xucleon .1 

It is this value which is significant in comparing the stability of 
two different isotopes. 

14.12 RELATIVITY AND SPACE TRAVEL 

Rockets for space exploration require highly efficient sources of 
thrust and large amounts of electric power. These requirements 
suggest nuclear power sources. In this sense, the mass-energy 
.■elation of relativity is important to space travel. But other pre- 
dictions of relativity, such as time dilatation, are probably not 
significant to space travel. 

If we could burn nuclear fuel so efficiently that one-tenth of 
the initial mass of the spaceship were converted into kinetic 
energy, the final speed would be less than 0.5c. This would give a 
very small (0.14) time dilatation — hardly enough to allow one 
generation of voyagers to reach destinations outside the solar 
system. 

PROBLEMS 

1. An atom moving at a speed of 1.0 X 10" m sec ejects an electron 
in the forward direction with a relative speed of 2.0 X 10" m/scc. Find 






202 Looking In: Atomic and Nuclear Physics 

the electron's speed as seen by an observer at rest (a) using a Newtonian 
transformation and (6) using a I.orentz transformation. 

Arts, (o) 3.0 X 10* m/sec, (6) 2.7 X 10 s m/sec 

2, Find the length of a meter stick when it is moving at a speed 
0.90c relative to the observer. Consider the cases when the stick is 
oriented («) parallel and (6) perpendicular to its direction of motion. 

Arts. («) 18.5 em, (b) 10(1 cm 

3. What speed will an electron have to acquire for its relativity 
mass to be twice its resi mass? Ans. 2.5 X 10 K m/sec 

1. What is (he energy equivalent of the mass of an electron? 

Ans. 0,51 Mot 

5. What is the radius of curvature of the path of an electron whose 
kinetic energy is 20 Mev when moving perpendicular to a magnetic 
induction of 0. 10 weber/rn-? Attn. 0.6S m 

h. Imagine that you are moving with a speed |c past a man who 
picks up a watch and then sets it down. If you observe thai be held the 
wmIcIi for (>,0 sec, how long does he think be held it? {Hint: You want 
It — U when you know t' t — l\.) Ans. 4.0 sec 

7. From the mass-energy relation, calculate the energy released in 
the reaction ,H= + ,H* - tHe*. (Data: ,H« = 2.014743 amu, Mr* 
= 4.003874 amu) Ans. 24 Mev 

8. A meson has a lifetime / = 1.0 X 10~*see before it decays. Find 
how far a meson with t> = 0.09c can travel. Ans. 300 m 

9. Find the energy liberated when an electron and a positron 
annihilate. Ans. 1.02 Mev 

10. If one uses the nonrehitivistic formula E,. = },m,r~. does one 
overestimate nr underestimate the kinetic energy of a particle of rest 
mass »io and speed vl 






15 

Hydrogen Atom 
Bohr Model 



... for the value of his study of the struc- 
ture of atoms and of the radiation emanat- 
ing from them. Nobel Prize citation 
for Niels Bohr, 1922 



By 1011, two rival pictures of the structure of an atom had 
evolved. J. J. Thomson suggested a "currant pudding" model of 
the atom iu which the positive charge was spread throughout a 
spherical volume of radius about 10 -ll> m, with electrons vibrating 
about fixed points within this sphere. Ernest Rutherford sug- 
gested a nuclear model of the atom in which the positive charge 
and almost the whole mass were concentrated in a very tiny 
nucleus; the electrons roamed through the rest of the atom, out 
to a radius of about 10~ 10 m. In crucial experiments, II. Geiger 
and E. Marsden probed the atoms in thin metallic foils with fast 
(" — sV c) a particles and showed that the observed deflections 
could be explained by the intense electric field near the center of 
a nuclear atom. Building on Rutherford's nuclear picture and 
using Planck's quantum hypothesis, X. Bohr fashioned a model 
of the hydrogen atom which explained its characteristic line 

203 






204 Looking In: Atomic and Nuclear Physics 

spectrum and correlated this with electrical measurements of 
excitation potentials and the ionization energy. 



15.1 NUCLEAR ATOM REVEALED BY ALPHA SCATTERING 

a particles are helium ions (He ++ ) and are emitted spontaneously 
by some radioactive substances. In the Geiger and Marsden 
apparatus (Fig. 15.1), a particles are directed against a thin 




Fig. 15.1 Apparatus for investigating rt particle scattering, showing: 
radioactive substance fi, the source of a particles, thin foil F of scattering 
material, zinc sulfide screen S, and microscope M. a particles emerge from 
a channel cut in the lead block I, strike foil F, and ore scattered to screen 
S. The conical bearing allows rotation of microscope and screen about ver- 
tical axis FF. [H. Geiger and E. Marsden, The London, Edinburgh, and Dublin 
Philosophical Magazine and Journal of Science, 25. 604 (I913).l 

metallic foil F in an evacuated chamber. The number of a par- 
ticles scattered at various angles with the original beam direction 
is found to decrease with increasing angle, but some a particles 
are scattered at angles greater than 90°, up to 180°. Rutherford 
found this "almost as incredible as if you had fired a 15-inch shell 
at a piece of tissue paper and it came back and hit you." For the 
IIe ++ ion is roughly 7,:i00 times the mass of an electron, and 
therefore the large deflections of a particles cannot occur by single 
collisions with electrons. Also, the foil used is so thin that a large 



Hydrogen Atom— Bohr Model 205 

a deflection cannot result from several successive collisions with 
electrons. But, Rutherford reasoned, if all the positive charge 
and most of the mass of an atom are concentrated in a very small 
nucleus, then the a particle can come very close to a large amount 
of charge all at once, and it will experience a large deflecting force. 
Further, since the mass of the deflecting nucleus is greater than 
that of the a particle, backscattering is possible. 

Rutherford derived an equation for a scattering based on the 
assumptions that the nucleus and a particle behave as point 




Fig. 15.2 

particles 
atoms. 



Deflection of a 
by nuclear-made! 



W^ 




Incident 
a particles 



Target 



positive charges, that Coulomb's law applies to the mutual repul- 
sion even at small distances, and that ordinary Newtonian 
mechanical principles hold (conservation of energy and conserva- 
tion of momentum). The number of a particles N reaching unit 
area of screen at distance r from the scattering foil was predicted 
to depend on 



r 1 4VW AV »i» 4 <»/2) 



(15.1) 



where A' a = initial kinetic energy of a particle 

N ( = number of a's incident per unit time on foil of thick- 
ness / having n target nuclei per unit volume 
Z = nuclear charge 
2e = a's charge 



206 Looking In: Atomic and Nuclear Physics 

In their precise and very readable report, Geigcr and Marsden 
neatly tabulated the results of counting thousands of a particles 
to show that .V was found to be proportional to (a) the thick- 
ness / of the scattering foil, (M the square of the nuclear charge 
Ze (using foils of Au, Ag, Cu, etc.), (c) the reciprocal of sin 4 
(0/2), where B is the angle of deflection, and ((f) the reciprocal 
of the square of the initial energy K a of the a particles (using 
different radioactive sources). 

The Geiger and Marsden experiments verified Rutherford's 
nuclear model of the structure of an atom. They clarified the 
meaning of the atomic number Z and showed it to be more sig- 
nificant than atomic mass in ordering elements in relation to 
chemical properties. An upper limit of 10 IS m was obtained for 
the size of the nucleus, in terms of distance of closest approach of 
a particles. The validity of Coulomb's law was verified down to 
about this distance of separation between charges. 

When Geigcr and Marsden used still more energetic a particles 
in their deflection experiments, some deviations from the scatter- 
ing pattern predicted by Kq. (15.1) were observed. This was the 
first hint of the existence of a "nuclear force" of attraction in 
addition to gravitational force and the electrostatic (Coulomb) 
force of repulsion. 

15.2 DATA FROM SPECTROSCOPY 

A grating spectrograph (Sec. 9.12) disperses the light incident on 
its entrance slit and focuses on a photographic plate a line image 
of the slit for each different wavelength present. As fine diffrac- 
tion gratings became available, owing largely to the skill of 
H. A. Rowland (1848 1901), spectroscopists diligently accumu- 
lated a vast number of measurements on the radiation emitted by 
atoms when excited in electrical discharge tubes, in ares, and 
otherwise. In general they found that (I) each element has its 
own characteristic line spectrum of wavelengths A or frequency v, 
(2) spectrum lines are generally sharp; elements producing the 
sharpest Hues are very stable; ('.',) lines in a sped rum may differ 
in relative intensity and in degree of polarization; (4) to the 
spectrum of every element can be ascribed a series of "term 
values" such that the frequency of every observed spectrum line 
can be obtained by differences of these term values. 



Hydrogen Atom— Bohr Model 207 

We shall consider the much-studied spectrum of hydrogen, 
the simplest atom. Its spectrum comprises several well-defined 
groups of lines: the Lyman series in the ultraviolet, the Maimer 
series in the visible region, the I'aschen series in the infrared, and 
others still farther oul in the infrared, Fig. 15.3. 

Balmer limit 



Visible p asc hen limit 
Lymon limit \ "■ *- j 

. '•« 1 1 i r i i .1 



-i_ 



j 



4,000 



8,000 



1 2,000 



16,000 



20,000 



Fig. 15.3 Some series of lines in the spectrum of hydrogen (wavelengths in 
Angstrom units). 

As a first step in developing au acceptable theory of atomic 
spectra, Rydberg found a relation which allows one to calculate 
the wavelengths in the hydrogen spectrum from differences 
between terms: 

l = R w ~ }y Khvve R = Lm x 10 ~ 3 A ~' < I5 ' 2 > 

When rtf = I and «,■ is given successive values, 2, 3, 4, 5, . . . , 
the differences of the terms in the Rydberg equation give the 
wave numbers 1 ,/X for the lines of the Lyman series. When n f = 2 
and it; = :>. !, ."», .... the Rydberg equation gives the wave 
numbers for lines of the Halmer series, etc. Although this formula 
was obtained empirically, it turns out to be closely related with 
the way the spectrum originates. 



15.3 BOHR'S THEORY 

There is a similarity of the hydrogen atom and our planetary 
system, in that in each case there is an attractive force inversely 
proportional to the square of the distance between the bodies. 
Bohr accepted Rutherford's concept of the nuclear atom and 
devised a model of the hydrogen atom in which orbital motion of 
the election was used to predict wavelengths of radiation which 
agree very closely with the observed wavelengths of the spec- 
trum lines (Table 15. 1), 



208 Looking In: Atomic and Nuclear Physics 

Table 15.1 Some term values end energy levels for hydrogen 

Wovenos. (1/cm) Joules 

* 



Elec. volts 













ty 












H H 


















t> 


O 


■~ 














N 


■nT 


cm 














r*> 


•o 


fo 














CN 


CM 


O 
































<0 


S 


PI 

in 














Batmer series 


u 


ti 


■j 


u 


^ 




00 


4 


i 

CM 


o 


8 
















CM 
00 


& 


d 


3 


s 

















3,047 
4,387 

6,855 
12,184 



27,419 



-6.0x10" 20 -0.38 



-8.7 
-13.6 

-24.2 



-54.3 



-0.54 
-0.85 

-1.51 



-3.39 



109,677 



-217.3 



■13.58 



man series 


6562A 
4861 
4341 
4102 


Sol 


tier series 


15, 233c 
20,264c 
23,032c 
24,373c 



3646 



(Limit) 27,419c 



The following assumptions are made in the Bohr theory of the 
hydrogen atom: (1) The electron moves around a stationary 
nucleus (a good approximation, since m, iue = 1,830m,.). (2) The 
electron is held in a stable circular orbit by the Coulomb attrac- 
tion between the negative electron and the positive nucleus. 
(3) Only certain (quantised) orbits are possible for the electron, 
namely, those for which its angular momentum is a whole-number 
multiple of h/2-ir, where h is Planck's constant. (4) Radiation is 
emitted (or absorbed) by the hydrogen atom only when the elec- 
tron undergoes an energy change in a transition from one orbit to 
another. The energy of the photon emitted (or absorbed) is given 
by 

hv - K, - /:, (15.3) 

Newton's laws of motion are assumed to lie applicable to the 
hydrogen atom, just as to bodies of larger dimensions. The force 












Hydrogen Atom Bohr Model 209 

of attraction exerted by the nucleus on the electron has the 
magnitude 



F = 



(l-,,.w- 



(15.4) 



The electron, moving with uniform circular motion, experiences a 
central acceleration a e = r 2 /r and a centripetal force mr- r, from 
Newton's second law. We equate the Coulomb force and the 
centripetal force 



r- 

(4xe u )r 2 



Hi !.' - 



(15.5) 



The kinetic energy of the electron is A* = £mr*. If, convention- 
ally, we take K p — when the electron is far from the nucleus, the 
potential energy /*.'„ of the electron in orbit is 



E = 
p (4irt«)r 

so its total energy is 



E = A* + K„ - Smv ! - 



(Smb> 



from which Eq. (15.5) gives 
I s" e» 



/; = 



2 (4jre«)r (-lvtn)r 



1 c 2 

2 (4Te )r 



(15.6) 



(15.7) 



(15.H) 



Of course, the kinetic energy of the electron is positive, but its 
total energy in a stable orbit is negative since it is bound to the 
nucleus, and work equal to |/i'| must be supplied to remove the 
electron from the atom (process of ionization), Fig. 15.4. 

Bohr's third assumption says that the permitted values of 
electron angular momentum are 



mm- = n rr n = ' i % 3, 

4/W 



(15.9) 



The radii of permitted orbits are obtained by solving Kqs. (15.5) 
and (15.9) for the quantity (mr) 2 : 

(mv)i - jEsp and (m » )s 






210 Looking In : Atomic and Nuclear Physics 



and equating the results to got 






- n' 



n _ 1, 2, 3, . . . 



4jr*me ! -l^me 4 

From Eq. (15.8), the total energy can he written 



(15.10) 



Sn = - 



I 



2 (4we D )r 



(4jreo)W 



n = 1, 2, 3, . . . (15.11) 

These are the only energy levels possible for the hydrogen atom in 
the Bohr theory. The energy values for levels I to arc indicated 





t 










Totcl energy 
E 




•:• UnquanHzed 




O 




Radius r — — 


__ f 




Tj 


i 




""■""-■■s^Binding energy 












r 3 


y 






l\' 




/ 






^^. , 


Ionization 


P= - — -*\ 






MM 


srgy 




1 + 











Fig. 15,4 Bohr's model of the hydrogen otom. 

in Table 15.1. The frequencies of radiation which the atom can 
emit or absorb are predicted from Eqs. (15.3) and (15.1 1) as 



A',- - E, 2tt'-W / 1 1 \ 



(15.12) 



Hydrogen Atom— Bohr Model 211 



Mini 



1 = " _ 2 * im < A (A L\ 

X c (4rco)%^e \n/ n, s / 



(15.13) 



li = 



= 1.097 X 10 s A-> 



from which the value of the Rydberg constant R can be verified as 
2xW 
(■iirtn)''/r'r 

The orbit for which n = 1 is referred to as the lowest state, 
the ground state, or the normal state for the hydrogen atom. If an 
electron in the lowest energy state receives 12.07 ev of energy by 
collision with an electron or by absorption of a photon, it can be 
"kicked up" into energy level n = 3 (see Table 15.1). The time 
interval before the electron spontaneously drops back to a lower 
energy level is called the lifetime of the excited energy state, and 
is ordinarily about 10 -s sec. The electron we. are considering 
might drop Brat from state n = 3 to « ■ 2, then from «. = 2 to 
n - 1. It would thus be responsible for the emission of two 
photons. One would have the frequency of the first (H„) line in 
the Balnier series; the other would contribute to the first line in 
the Lyman series. 

The ionization potential is defined as the energy needed to 
remove, from an atom an electron initially in the lowest energy 
state. The lesser energy needed to promote an electron from one 
state to another of greater energy is called an excitation potential. 
Obviously the hydrogen atom lias only one ionization potential, 
but several excitation potentials. An atom with many electrons 
lias a corresponding number of ionization potentials. Because of 
the Pauli exclusion principle (Chap. 10), only in II and He do all 
the electrons have it = I in the ground state. In other atoms, the 
ground state is taken as the state of lowest energy. 



15.4 EXTENSION TO HYDROGENLIKE IONS 

Bohr's model and theory apply successfully to ions which have 
only one electron, that is, I!e + , Li ++ , Be s+ , etc. The equation for 
the Coulomb force is modified to read F = e(Ze) / (■iwuijr* where 
Ze is the charge on the nucleus. This leads to inclusion of Z s in 
Eq. (15.13), and thus 

I-W-L -JL\ (15.14) 

X \nr my 



212 Looking In: Atomic and Nuclear Physios 



Hydrogen Atom— Bohr Model 213 



This relation predicts that He + (Z = 2) should radiate a series of 
iiiies hi the visible region for transitions to «/ = 4 similar to the 
Balmer series for H (Fig. 15.5). This Pickering series for Hc + was 



He* 



*t 



■4 - 
■3 - 



— 8: 

— 6- 



Fig. 15.5 Comparison of energy 
levels for H and for He 1 . 



n = \ 



n = 2 



observed first in star spectra and was subsequently identified witii 
a laboratory helium light source. 

15.5 CORRECTION FOR CENTER-OF-MASS ROTATION 

The frequencies in the Pickering series for He + are not precisely 
the same as those in the Maimer series for H, as Bq. (15.14) pre- 
dicts. Also, the heavy isotope II* has spectrum lines slightly 
shifted in frequency from those which Kq. (la. t:i) predicts should 



M 



CM 



Fig. 15,6 Rotation about center of mass (CM). 

be identical for both H 1 and TI-. These discrepancies suggest thai 
instead of simply considering that llic electron moves around a 
fixed nucleus, we should consider that both electron and nucleus 
move about their common center of mass (Fig. 15.6). Let r f and 
r„ be the distances from the center of mass to the electron and to 
the nucleus, respectively. Then r = r e + r„ is the distance. 






between electron and nucleus. If we introduce the angular 
velocity w - »/r, [Kq. (15.5)1, for the centripetal force 



mi- 



(4r tu )r : 



becomes nr \ « = mr *>° (18.15) 

t-Mreojr- 



The equation quantizing angular momentum, Eq. (15.9), becomes 



mr,*a> + .l/r„ 2 u = n 7 r- 
2ir 



(15.16) 



where .1/ is the mass of the nucleus. From the definition of center 
of mass, 

M 



m + ,1/ 



and 



r„ = 



m 



m + M 
By combining Eqs. (15.15) and (15.17), we find 



(4^jH = m ">* rwi 



and 



!•'*,.< = 



w m irw ■"fts- 

2.TT 

where the symbol »i r ..,i is used for the "reduced mass," 
m\! m 



Wr,.i = ; 



m + .1/ 1 + tn/M 



(15.17) 



(15.18) 



( 1 5. fit) 



< 1 5. JO. 



Since Fqs. (15.18) and (15.19) differ from Eqs, (15.15) and (15.16) 

for no nuclear motion only in the replacement of electron mass m 
by reduced mass »w, we see that the energy /■.'. corrected for 
motion of the nucleus is related to the uncorrected energy A'„ of 
Eq. (15.11) by 



B. = m "" /■:„ - 

m 



E„ 



1 + w M 



(»-«)* 



(15.21) 



since m/M « 1 . 

If we use the corrected expression for energy in Eq. (15.11), 
the Bohr equation (15.13) for wave numbers becomes 



1 u _2irV_ mM f \_ I \ 
X (47r e „} a A 3 c m + M \rf nf) 



(15.22) 






214 Looking in: Atomic and Nuclear Physics 

This correction shifts each energy level by ahout 0.055 per cent 
for H 1 . For the isotope II 1 , the shift is less. Hence a frequency 
difference can be observed when two isotopes of an element are 
present in a light source. The first (II a ) line hi the Balmer series 
of II 1 has wavelength (in(>2.80 A; that for II 2 lias wavelength 
6561.01 A. The reduced mass correction also explains why the 
energy difference, and hence the frequency of radiation, is slightly 
greater for the helium ion He + (say, from n = 6 to n = 4) than 
for the corresponding transitions (n = 3 to n = 2) for hydrogen. 

15.6 THE CORRESPONDENCE PRINCIPLE 

Bohr's correspondence principle is the guiding idea that, in the 
limit, the laws of quantum theory must join and agree with 
classical theory (which does not involve It). This asymptotic 
approach is to be expected when we go from microscopic systems 
to those of larger dimensions, or for large values of the quantum 
number n. 

Bohr's theory for the hydrogen atom does show such agree- 
ment. With the aid of Kqs. (15.5) and (15. 10), we may express the 
frequency of rotation of an electron in a Bohr orbit as 



f'orb — tj-~ — 



v 
2Hr 



I 



C4 



ire u )mr 



■]' = 



me* 



WllW 



(15.23) 



On classical (Maxwell) theory we should expect this electron to 
radiate energy of this frequency, and possibly its harmonics. 

But the theory which includes Bohr's quantum assumptions 
for the 11 atom gives for the frequency radiated 



me* (I _1_\ 



Now 



ii/ 2 



nr 



-nf 



«,' 



«,-%/ 



(n, 4- n,)(nj - n f ) 



If rii and n; are both large compared to 1 and if An is small, we 
can write this approximation 



nf 1 



n, s 



2n Am 



2An 
n 8 



Hydrogen Atom — Bohr Model 215 

where An = tii ~ n/ and n»?ij« %. Then the Bohr frequency 
becomes 



Vln.lir ~ 



me* 



4« 2 A s a 3 



An 



(15.24) 



Comparison of Eqs. (15.24) and (15.23) shows that for large 
orbits (large n) and for Are = 1 , the atom radiates the frequency 
expected from classical electromagnetic theory. For An > 1, we 
get harmonics. This is an example of a transition region between 
macroscopic and microscopic physics where the laws of classical 
physics and quantum physics overlap. 

QUESTIONS AND PROBLEMS 

1. Which of the experimental observations mentioned in Sec. 15.2 
are satisfactorily explained by the Bohr theory of the H atom? Are any 
not explained? 

2. How can the fact that the spectrum lines of hydrogen are sharp 
be used to support the statement that all electrons have identically the 
same charge ef 

3. At what temperature will the mean kinetic energy of hydrogen 
atoms be just sufficient to excite the H a line? Am. 93,40G°K 

4. Selig Hecht showed experimentally that a dark-adapted human 
eye experiences the sensation of light when the retina is irradiated by as 
little as 10 X 10~ 12 erg. What is the minimum number of quanta of 
yellow light (5,893 A) which the eye can detect? Arts, about 3 

5. Assume that a free electron having kinetic energy 24.2 X I0 -so 
joule unites with a H + ion, goes to the lowest (n = 1) level, and gives 
up its energy in a single photon. What is the frequency of the photon 
radiated? Ans. 36 X 10'Vsec 

6. How much energy is there in a quantum of violet light, wave- 
length 4,358 A? In a quantum of yellow light, wavelength 5,893 A? 

Ans. 2.84 ev, 2.10 ev 






16 

Quantum 
Dynamics 



In this paper 1 am going to attempt to 
find the foundation for a mechanics of 
quantum theory. This mechanics is 
based exclusively on relations be- 
tween quantities which are observable 
in principle (e.g., frequencies and in- 
tensities of line spectra, and not elec- 
tron orbits), ... W, Heisenberg, 
1925 



Beginning with Bohr's initial formulation of the quantum theory 
of atomic structure in 1914. physicists recognized that the 
mechanics of systems of atomic dimensions must obey laws differ- 
ent- from the larger systems successfully described by the classical 
mechanics of Newton. By 11)24, a new method of treating atomic 
phenomena began to be developed. It is known as quantum 
mechanics, quantum dynamics, or wave mechanics. The names of 
L. de Broglie, K. Sell nidi tiger. W. Heisenberg, P. A. M. Dime, 
ami EL U. Condon arc chiefly associated with this development. 
The concepts discussed in this chapter bring us to an acecpt- 

216 






Quantum Dynamics 217 

able theory of atomic physics. We arrive at a logical branching 
point in our path. Armed with a successfully tested theory of the 
atom, we can now (I) try to understand and predict properties of 
atoms in intimate aggregation (solid-state physics) or (2) we can 
turn to investigation of the internal structure of atoms. A goal of 
such nuclear studies might be ultimately to manipulate nuclear 
particles to our use, as a chemist manipulates atoms to create 
molecules with desired proper! tee. 

16.1 PARTICLES AND WAVES 

Planck's derivation of the law for the energy distribution of 
blackbody radiation (1900) first brought to light the particle 
(quantum) aspect of electromagnetic radiation. Einstein strik- 
ingly established this viewpoint with his explanation of the photo- 
electric emission of electrons from solids (1905). Photons were 
endowed with momentum {hv/c) by the Conipton effect (1924). 
Also in 1924, Louis de Broglie, proceeding from relativity theory 
and the observation that nature is symmetrical in many ways, 
suggested that whenever there are particles witli momentum p, 
their motion is associated with (or "guided by") a wave of 
wavelength 



V 



(16.1) 



The square of the amplitude of the de Broglie (matlcr) wave in a 
given region is interpreted as being proportional to the prob- 
ability of finding the particle of momentum p in that region. In 
de Broglie's hypothesis about wave-particle duality, an electro- 
magnetic wave tS the de Broglie wave for a photon, and proceeds 
with speed c. The de Broglie waves for electrons, protons, neu- 
trons, etc., are That electromagnetic waves, but "matter waves," 
which travel with the speed of the particle. 

We shall now discuss (I) a verification of these de Broglie 
waves and (2) something about how their value at various 
points in space may lie calculated. 

Since the de Broglie equation predicted that 100-ev electrons 
should have wavelengths of about I A, it was suggested that the 
wave nature of mailer might be tested in the same way that the 
wave nature of x rays was first tested. A beam of electrons of 



218 Looking In: Atomic and Nuclear Physics 

appropriate energy could be directed onto a crystalline solid 
(Fig. Hi. In). The atoms of the crystal form a three-dimensional 
array of diffracting ©enters for the de Broglie wave guiding the 
electrons. There should be strong diffraction of electrons in 
certain directions just as for the Bragg diffraction of x rays. 






(a) 




O 





(c) 

Fig. 16.1 (o) Davis son and Germer apparatus, {fa) Angular 
distribution of secondary electrons, (c) Interpretation in terms 
of Bragg reflection of electrons (refraction of rays has been 
omitted). 



This idea was tested by C. J. Davisson and L II. Germer 
using 54-ev electrons and a crystal of nickel (Kig. HUfe). The 
emerge] i ( beam showed an intensify peak for 6 = 50°. The wave- 
length calculated From the Bragg equation turns out to be just 
h/p for a 54-ev electron. (The fact that electrons are observed at 
other angles is attributed to secondary emission: Some incident 



Quantum Dynamics 219 

electrons collide, with and share their kinetic energy with some of 
the electrons in the solid, with the result that some of these are 
emitted at random angles.) Experiments on electron diffraction 
confirms the hypothesis that their motion is directed by a wave of 
some kind, and the wavelength agrees with that predicted by the 
de Broglie relation, A = h/p. 



16.2 DIFFRACTION OF PHOTONS AND NEUTRONS 

De Bro^lie's hypothesis suggests that particles of any type may 
exhibit diffraction effects. The diffraction of neutrons has been 
useful in the investigation of the structure of solids. Beams of 
neutrons whose wavelength is roughly equal to the spacing of 
atoms in a solid can be obtained from a nuclear reactor. These 
beams are diffracted by layers of atomic nuclei. On the other 
hand, x rays are diffracted from planes in the solid where the 
density of electrons is highest. Thus the two types of experiment 
can give supplementary information about the structure of a 
solid. X-ray investigations reveal the location of the (bound) 
electrons in a solid ; neutron diffraction reveals the arrangement of 
the nuclei. 



16.3 WAVE MECHANICS 

We expect that a de Broglie wave will obey the same type of 
second-order differentia] equation (Appendix C) used to represent 
other waves (Chaps. 9 and 12). 

Important applications of Schrodiugor's equation are to cases 
where the electron is subject to forces which hold it in a certain 
region, as in an atom or in the atomic lattice of a metal. The 
potential energy of the electron then varies from point to point. 
As the simplest case of this type, let us examine the wave function 
i>(x,i) for a particle of mass m which can move along a line 
between stops a distance L apart, like a bead on a stretched wire. 



Fig, 16.2 Particle confined to 
linear motion within range L. 



m 

-o 



220 Looking In: Atomic and Nuclear Physics 

The particle will never be outside the interval < x < L; so 
^ is zero for x < mid x > L. Inside the region considered f here 
are no forces on the particle; it is a free, particle. So the wave 
equation has sine and cosine, solutions, hut these must he zero at 
the. ends of the allowed interval. The allowed wavelengths of the 
de Broglie wave arc A = 2L/n, which leads to 



h nh 
P = \ = 2L 



(lfi.2) 



showing that the linear momentum is quantized. The kinetic 
energy of the particle is 



p a I ii' ; h- 
k ~ 2m~ 2m 4JS 



8mP 



(18.3) 



and since we have taken K v = 0, the total energy must have one 

of flic values 






The particle is located by the matter wave 

iff = ( ^D sin - . J cos ait n = 1 , 2, 3, , . . 



(16-4) 



(16.5) 



The amplitudes of the standing waves for states of motion cor- 
responding to n = 1, 2, 3, ... , vary as shown in fig. 1(5.3. 
(There is a close analogy with standing waves in a vibrating 
string.) We see that the act of localizing or hounding a particle 
leads to the requirements that (1) the energy of the system can 
take on only certain values and (2) zero is not a possible value of 
the kinetic energy. 

Another important type of prediction from wave mechanics 
deals with the "leakage" of particles across an energy barrier. 
Suppose we have a particle bound in a shallow potential energy 
"hole" (1'ig. Hi. 4). There are now two kinds of solutions for the 
wave equation. There are solutions for any K > 0. Particles in 
these states have enough energy to escape; ^ extends over all 
space for them. Hut for particles whose lit is less than #„n (Tig. 
1G.4), the total energy is negative, and for K < the wave equa- 



Quantum Dynamics 221 

tion has solutions for only certain values of the energy. The 
higher the energy the more nodes there are in the wave (Fig. 
16.46). The solutions are sinusoidal inside the well and have 
exponential tails outside. Thus (here is some probability for lind- 



Fig. 16,3 Wave functions for a bead on 
a string, for states n = 1,2,3,4. 




'1 4 








x— *- 


'. 


t 








>■■ 








'>i 




(a) 





X-— \L 




Fig. 16.4 (a) A "square" potential hole, and (b) the wave function of its states 
n = 1,2,3. 



ing particles in a region where, according to classical theory, they 
do not have enough energy to be. Around a nucleus we may think 
of a potential harrier whose craterlike shape is determined by the 
Coulomb electrostatic force and a shorter-range 1 force of nuclear 



222 Looking In: Atomic and Nucfear Physics 

attraction. The wave viewpoint predicts that, charged particles 
which do not have enough energy to go over the top of this 
barrier have a small but not zero probability of occasionally 
tunneling through the barrier. 



16.4 BOHR ORBITS OR DE BROGLIE WAVES? 

If we apply the concept of matter waves and the probability 
interpretation of Sehrodinger's equation to the hydrogen atom, 
we find that I he features which the Bohr theory correctly pre- 
dicted (only with the aid of arbitrary assumptions: mr = nh/'lir, 
etc.) follow as a natural outcome of the mathematics involved. 
The quantum dynamical treatment provides additional informa- 
tion as well. The electron in a hydrogen atom has potential 
energy -(■'■ \irt„r. If we write for the radial distance 



r - V-t 2 + y* + z 1 
Sehrodinger's relation |Appendix C, Kq. (4}| becomes 

(16.fi) 

The solution of this equation, ^(r,ij,z), is a function which has a 
definite value at each point in the neighborhood of the nucleus. 
To discuss this equation, it is convenient to change from reetan- 




Fig. 16.5 Rectangular and spherical co- 
ordinates. 



Quantum Dynamics 223 

gular coordinates to spherical polar coordinates, using the 
relations 

r = distance of point a from origin = y/x 2 + y* + z z 

8 = angle from z axis to r = cos~ l {z/r) 

4> = angle around z axis measured from x axis = tan -1 (</ as) 

With the introduction of the coordinates r, Q, $, Kq. (Ifi.O) can be 
separated into three ordinary differential equations, a fact we 
represent by 



f(r,e,<t>) = rt(r)0(0)*W 



(lfi.7) 



The function li describes how f varies as we go out from the 
nucleus in a definite direction. The functions 8 and <$ describe 
how ^ behaves from point to point on a sphere of radius r. 

The equation for the function K(r) has a solution for any posi- 
tive value of E. These solutions correspond to states in which 
the electron has enough energy to escape from the atom; there 
are no quantum restrictions on the energy of a free electron. But 
there are only certain negative values of E for which Kq. (16.7) 
has any continuous solution. When the electron is bound in the 
atom, an acceptable wave function ^ exists only if E has one of 
the particular values 



E„ = 



-me 4 1 



-I3.fi 



ev 



n = 1, 2, 3, 



(10.8) 



These are the same values for the energy states that the Bohr 
theory predicted. The quantum number n is here related to the 
part R(r) of the wave function which describes the probability 
per unit volume of finding the electron in a given volume element 
at various distances from the nucleus. This is independent of 8 
and *. We can compute the average distance of the electron from 
the nucleus by averaging over the probability distribution. The 
result is roughly the same as the radius of the first Bohr orbit. 
The energy (Kq. 10.8) is in exact agreement with the Bohr theory. 
For each value of n, the equation for 8(8) is found to have one 
or more solutions, described by a second quantum number I. 
This quantum number takes on only the values 



/ = 0, 1,2,3, 



n - 1 



(16.9) 



224 Looking In: Atomic and Nuclear Physics 

Solutions of the * equation arc related to solutions of the R equa- 
tion such that the electron is less likely to he found near the 
nucleus when in a high-/ state than when in a low-/ state of the 
same energy. 

For each value of /, the equation for *(<£) is found to have one 
or more solutions, designated hy a third quantum number jtcj. 
This takes on only the values 



m , = -/,_(/- i), -(I -2), 



— 10 12 

(I - I), I (16.10) 



Xo solutions of Schrodinger's equation for the hydrogen atom 
exist for any other values of «, /, and m ( . 

16.5 THE QUANTUM-NUMBERS GAME 

An atom can be completely described by the use of just four 
quantum numbers for each electron. Three of these we have 
already introduced. 

The principal quantum number » determines the energy, 
E<|. (10.8). It may have the integral values n = 1, 2, :i, . . . . 

The orbital angular- momentum quantum number I deter- 
mines the angular momentum of the motion of the electron about 
the nucleus. It may take on any integral value from to n — 1. 
The corresponding value of the electron angular momentum i- 
VKI+ 1) h/2n. 

The component of the orbital angular momentum along the 
a axis is given by null, 2x, where m may lake on any of the 2/ -j- I 
values: 0, ±1, ±2, . . . , ±1. The quantum number mt is called 
the magnetic quantum number because physically the presence, 
of an external magnetic field is necessary to establish a reference 
direction (z axis) in space. In a magnetic field, the electron's 
angular momentum is said to be "space-quantized" because its 

component along the di recti >i the magnetic field is restricted 

to the values mji 2tt (Fig. l(i.(i). 

We now introduce a fourth quantum number 8, the electron- 
spin angular-momentum quantum number. This quantum num- 
ber defines the internal angular momentum {and associated mag- 
netic moment) which an electron is found to have, independent of 
its orbital motion. An experiment to show this property of the 
electron was performed by Stern and Clerlach. If a neutral atom 






Quantum Dynamics 225 

which has a magnetic moment passes through a uniform magnetic 
field, it experiences a torque, but no deflecting force. If, however, 
the field is nonuniform, the atom experiences a net deflecting 
force as well. Consider a beam of II atoms. The electron in the 
normal state has zero orbital angular momentum for n = 1, 
/ = 0, mi = 0. There is no magnetic moment due to orbital 
motion. Vet the beam of II atoms is observed to split into two 
parts, each associated with a restricted orientation of the clec- 




Fig. 16.6 Possible orientations of angular-momentum 
vectors. 

tron's spin angular momentum. The two possible values of the 
component of the spin angular momentum in the direction of the 
magnetic field are ±%k/2*. We conclude that, unlike the other 
quantum numbers, which are integers, n can have only the value 
£, The component of the spin angular momentum may be either 
parallel or antiparallel with the applied magnetic held. So we can 
define a spin magnetic quantum number m„ - ±\ and write the 
component of the spin angular momentum in the direction of the 
applied field as mji/2w. 



(«> 



J Traces on 
receiving 




I 

Direction atoms move 1 



Fig. 16.7 Magnetic field used in Stern-Gerloch experi- 
ment, (a) With no field there is a single beam, (b) With 
field, beam splits; some atoms are deflected toward N po!e, 
some toward S pole. Traces where beam strikes detecting 
plate are shown at top. (Adopted from R. 0. Rusk, "Afomk 
and Nuclear Physics," Applet on- Century -Crofts, Inc., New 
York, 1958.) 



Quantum Dynamics 227 

16.6 THE PAULI EXCLUSION PRINCIPLE 

In 1925 W. Paul! suggested that a complete description of the 
atom must include a unique description of each electron in the 
atom. Xo two electrons in an atom may have identical values for 
a set of four quantum numbers. 

To see how this rule operates, consider the number of elec- 
trons permitted in the first orbital group (or shell) for which 
n = 1. Since n is 1, 1 = and m t = 0, But m, may be +1 or - '.. 
So in this first group there may be two electrons, distinguished 
only by having their spins in opposite directions. The continua- 
tion of this assigning of quantum numbers to electrons in many- 
electron atoms is shown in Table 16.1. 

Table 16.1 Numbers of electrons in groups (or shells) as determined by Pauli's 
exclusion principle 



Orbital 
group 




n 


/ 


m 


s 


No. elec. in No. elec. in 
subgroup completed 
group 


1 


{ 


1 
1 











i 


2s 1 2 




Iii the terminology of Table 16.1, we replace the term "orbit" 
by "group" or "shell" (determined by n). This emphasizes the 
three-dimensional nature of the atom. The shells are often named 
the K, L, M, . . , ,Q shells, corresponding ton =» 1,2,3, ... , 7. 






226 



228 Looking In: Atomic and Nuclear Physics 

Within a shell, electrons with a common value of / form a sub- 
shell These arc designated s, p, rl, or / subshells according to 
whether / has the value 0, 1,2, or 3, 



16.7 BUILDING THE PERIODIC TABLE OF ELEMENTS 

When the elements are arranged in order of increasing atomic 
number, a periodicity in their chemical properties becomes 
apparent, as shown by Mendeleev. The structure of the periodic 
table is in agreement with the ideas of filled shells and subshells 
as predicted by the 1'auli principle. We may "build up" an atom 
by putting each electron in the shell of lowest energy until the 
quota of permitted states is filled. Any additional elections must- 
be put. in the next shell as shown in Table lfi.2. The final column 



of Table KS.2 is a description of the electrons in the outside shell 
for the normal (ground) state of the atom. The electron configura- 
tion of an atom is described by the abbreviated not at inn ui" t he 
last column. For example, %- means there are two electrons in 
the n - .i, I = I subshell. 

The quantum numbers we are using were originated for the 
case of one electron. J I is remarkable thai by assi g nin g occupied 
states in terms of these numbers we get ati accurate description of 
many of the properties of complex atoms. Kvidently the various 



Table 


16.2 


Electron configured 


ion for 1 


ght atoms 










>»,/-* 




Z 


1,0 


2,0 


2,1 


3,0 


3,1 


3,2 


4,0 


Configu- 


Element 




(U) 


(2s) 


(2 P ) 


(3*) 


(3 P ) 


(3d, 


(4s) 


ration 




H 




1 


1 














If 


He 




2 


2 














w 




Li 




3 


2 


I 












2* 




Be 




4 


2 


2 












2s* 




B 




5 


2 


2 


1 










2p 




L 




6 


2 


2 


2 










2p= 




N 




/ 


2 


2 


3 










2p* 




O 




8 


2 


2 


4 










2p 4 




F 




9 


2 


2 


5 










2p» 




Ne 




10 


2 


2 


6 










2p r> 




No 




11 


2 


2 


6 


T 








3s 





Quantum Dynamics 229 

electrons in a complex atom must disturb each other's orbits very 
little. 

One sort of disturbance, called screening, should be men- 
tioned. An outer electron is in a weak electric field because inner 
electrons screen it from the positive charge of the nucleus. Hence 
states in which the electron has some probability of being found 
very near the nucleus will have lower energy (greater binding) 
than those states in which the electron tends to stay outside the 
screening inner electrons. Of the solutions of the Schrodinger 
equation for a given n, those with lower values of J will tend to 



/ 

2.1 

16 
12 

B 

4 - 




_i I 1 I i i i 



■ !■■! 



I ... I 



10 



I i 1 



14 



IS 



22 



26 



30 X 



Fig. 16.8 Variation of ionization energy (J in ev) with atomic number Z, sug- 
gesting greater stability of certain electron configurations. 



penetrate the cloud of screening electrons most. Hence, for atoms 
containing more than one electron, penetration causes the energy 
of an orbit to depend on I. as well as on ti. (In terms of the liohr 
picture, energy depends on the shape of the orbit as well as on its 
size.) 

Klectrical measurements which correlate well with electron 
configurations are shown in I'ig. Ili.8. The ionization energy is 
the work needed to remove the least tightly bound electron from 
an atom. The variation of ionization energy with atomic number 
7. suggests that certain electron configurations have relatively 
great stability. The first is for helium, where the it = I shell has 



230 Looking In: Atomic and Nuclear Physics 

its ii"" 1 '' 1 <>f two electrons. The sharp drop to the binding energy 
for lithium is attributed to the fact that the third electron must 
bi> ndded io l lie n - 2 shell and is therefore farther from the 
nucleus. Tor the elements after lithium, there is a trend toward 
increasing binding energy until another maximum is reached at 
neon, when the n = 2 shell is filled. Like He, Xe is an inert gas. 
This variation in binding energy is repeated several times in the 
periodic table, each time giving a maximum binding at an inert 
gas, followed by a minimum for the succeeding alkali metal. The 
size of atoms also oscillates from shell to shell, about a value 
approximately 1 A for the radius. In each shell, the alkali metal 
has the largest radius. 

16.8 CHARACTERISTIC X-RAY SPECTRA 

When a target is bombarded with electrons of high energy (l>), 
x rays are produced which have a spectrum which is continuous 
up to the maximum frequency given by the relation fcjWx = I <■■ 
In addition, x-ray spectrum lines arc observed at frequencies 
which are characteristic of (determined by) the target material. 
Characteristic x-ray spectra can now be explained in terms of the 
shell structure of atoms. First, a vacancy must he created by the 
displacement of an inner electron from, say, the K or L shell. 
Since there are usually no near-lying vacant energy levels to 
which these electrons may be promoted, they must be removed 
altogether from the atom (ionization). This may he accomplished 
when atoms of the target are bombarded by electrons which have 
been accelerated through a potential difference of many thousand 
electron volts or by high-frequency photons. Transition of a near- 
lying electron then occurs to Jill the vacancy. If a vacancy in the 
K shell is filled by an electron from the h shell, an x-ray photon is 
radiated whose frequency depends on the difference in energy 
between the K and L shells. The vacancy left in the L shell is in 
turn tilled by an electron from a still higher energy state, with 
radiation of a photon of somewhat lower frequency, Fig. lfi.9. 

Since the energy of the electron in the K shell is chiefly 
determined by the nuclear charge Z, Moseley found he could use 
the K a lines of the elements to identify the atoms in the target of 
the x-ray tube. lie found a linear relation between the square 
root of the frequency and (Z — l), as would be expected from 



Quantum Dynamics 231 

the Bohr formula with allowance for screening by the inner K 
electrons. From the relation 



= Rc(z - ]y-(L- L\ 



(16.11) 



Moseley was able to prove that early assignment of atomic 
numbers to cobalt and nickel was in error. The atomic mass of a 
natural mixture of the isotopes of Xi is 58.(i!) and for Co 58.94. 
They were first placed in the periodic table in the order of increas- 
ing atomic mass. Hut the x-ray lines showed that this order 
should be reversed, for Z Co = 27 and Zy, = 28. Moseley 's work 



K 




0.2 0.4 0.6 0.8 1.0 
Wavelength, A 

(a) 



E, 



E M 
(b) 



K« 




K » 








L a 




H 






\ <M a 



M 
N 



Fig. 16.9 Characteristic x-ray spectrum, (a) Molybdenum target with V 
(a) Simplified energy level diagram. 



35 kv. 



(1913) gave the first accurate method for measuring atomic 
number, Z. The committee awarding the 1917 Xohel Prize to 
C. G. Barkla, for his work on characteristic x rays, stated that 
Moseley would have shared the award but for his death at 
Callipoli. 

16.9 PHYSICS OF THE SOLID STATE 

Our theory, based on the nuclear atom model and quantum 
mechanics, tells us that under ordinary circumstances of tem- 
perature and pressure the nuclei of atoms will never get very close 
to one another. The combination of atoms should therefore be 






232 Looking In: Atomic and Nuclear Physics 

explainable through the exchange or sharing of electrons. In 
terms of tfic measured masses and charges it should be possible 
to describe the formation of molecules and chemical reactions. 
One might also hope to describe crystal lattices and the mechan- 
ical, thermal, electric, and magnetic properties of solids. Prac- 
tically, the difficulty is the complexity of the computations. We 
shall examine some of the successes of quantum mechanics in 
explaining important electric and magnetic properties of solids. 
This comprises hut one segment of solid-state physics in which 
there is very active research. 



16.10 CLASSICAL THEORY OF CONDUCTION IN METALS 

A theory proposed by Drude and Lorentz, soon after the dis- 
covery of the election, assumed, as have later theories, that some 
of the electrons are free to travel throughout the whole volume of 
a crystalline material. In a "good" metal, it was assumed thai 
there is about one free electron per atom and that the number of 
conduction electrons is independent of temperature. These elec- 
trons dart around in all directions ui) h the high speeds of thermal 
agitation. Hut if an electric held is applied, the "electron atmos- 
phere" experiences a relatively slow drift, superposed on the 
random thermal motions. The electron drift is the electric current. 
The transfer of any increase in the energy of random motion in 
any direction constitutes thermal conduction. To make quantita- 
tive predictions, it is necessary to make some assumptions about 
the distribution of electron speeds. Theories have differed in 
these assumptions. 

The classical theory assumed that the electron speeds followed 
the same distribution law as .Maxwell and Bollzmanii had used 
for molecular speeds in developing a successful kinetic theory of 
gases (Chap. 8). Among a large number N of electrons, the frac- 
tional number N,'N having speed r is given by 



AT vV \2fc27 



(16.12) 



Tf we plot this expression against o, the area under the curve 
between r, and v-> equals the fraction of all the electrons whose 
speeds are between r, and r*. Since kinetic energy depends on the 



Quantum Dynamics 233 

speed squared, the average kinetic energy depends on the average 
of the squares of the speeds. The square root of this average is 
called the root-niean-square (rms) speed. The distribution curve 
becomes flatter and the maximum shifts toward higher speeds as 
the temperature increases (Chap. 8). 

The classical theory gives rough predictions of the electrical 
and thermal conductivities of metals, if is in accord with the 
experimental observation that the best conductors of electricity 
are also the best conductors of heat. Wiedemann and Franz ( 1 850} 
showed that the electrical conductivity thermal conductivity 
ratio is a constant, for metals. The classical theory, using known 
values for e and k, predicts that the thermal conductivity/ 
electrical conductivity ratio = li.ll X 10 _B 'T cal ohm sec (°K). 



Fig. 16.10 Hall effeel 



D D' 
—n — 



This checks well with values measured for platinum and other 
I mre metals. 

But the classical theory meets with significant failures. It 
predicts that the free electrons should contribute |/f to the 
specific heat of a crystal. This considerable electronic specific heat- 
is not observed experimentally. Also, the theory is unable to 
explain the enormous range of electrical resistivity for different 
materials. Further, the theory suggests that since the free elec- 
trons have magnetic moments, even a weak magnetic field should 
produce a large paramagnetic magnetization (magnetic moment 
per unit volume) in a conductor. It does not. Finally, the theory 
has difficulty in predicting the sign of the Hall coefficient. For a 
current-carrying conductor (Fig. Hi. 10) one would expect that a 
potentiometer connected between Cand J), in a plane perpendicu- 
lar to the current . would indicate zero potential difference. If now 
an external field H is applied, the conduction electrons experience 
a magnetic thrust perpendicular both to H and their velocity v. 
The equipotential line CD is tilted through some angle <j> to 
position CD'. The classical theory predicts that tan 4> (the Hall 
coefficient) should have the same sign for all metals. It docs not. 



234 Looking In: Atomic and Nuclear Physics 

16.11 FREE-ELECTRON QUANTUM THEORY OF CONDUCTION 

Fermi introduced a radically different description of the free 
electrons iii a metal. He incorporated the exclusion principle, 
assuming that the "free" electrons hi a metal are quantized and 
that no two can act exactly alike. Momenta are quantized; only 
two electrons (having opposite spins} can have a given mo- 
mentum. As the temperature is lowered, electrons settle down hy 
quantised slops to lower momentum values. But as a consequence 
of the exclusion principle, some electrons wil! remain at mo- 
mentum values considerably above zero: thai is, linn- will have 
appreciable energy, even at absolute zero temperature. When the 



N 



f 


J0°K 
i 




300° K 




\Very high temp. 



Fig. 16.11 Fermi distribution of speeds at various 
temperatures. 



temperature rises, only the electrons of highest momentum can 
accept thermal energy and move to still higher momentum 
values. 

The Fermi distribution law is expressed by 



N = h l e m, "' i!_A "-'* r 



1 



(16.13) 



where E,„ is the maximum energy an electron can have at 0°K. In 
Fig. 16.11, the progressive rounding of the curve as temperature 
increases represents the shift of some electrons to higher energies. 
The Fermi distribution curve should be compared with the Max- 
well distribution (Chap. 8). 

The ['ermi theory successfully accounts for the slight partici- 
pation of electrons in specific heats. In Fig. Hi. 12, the Fermi 
distribution of energy is plotted. At 0°K all energy states are 



Quantum Dynamics 235 

occupied up to a certain maximum (Fig. 16.12a). At a higher 
temperature some electrons in upper levels have been able to 
accept energy ami move to still higher [ovate (Kg. [6.126). Btl1 
owing to quantum restrictions, relatively few electrons have 
participated in the temperature rise. The Fermi theory predicts 
that electrons in a conductor should contribute roughly 1 per cent 
of the amount predicted by the Maxwell theory, in agreement 
with experiments in calorimetry. 

The fact that all energy levels, up to a certain maximum, are 
filled means that for every electron traveling to the right in a 
metal there is another elect ion traveling toward the left. Thus all 
electrical conduction in the metal must be due to the relatively 
few electrons near the top of the distribution (Fig. JO. 126) which 



Fig. 16.12 Fermi distribution of 
energies, showing (a) all levels filled 
up to a maximum of K, (b) some 
electrons promoted to higher energy 
levels at a high temperature. 




Rel, no 




Rel * no 



can be excited easily to an unoccupied quantum level. One con- 
cludes that electricity must be conducted by only a small fraction 
of the free electrons (rather than by all, as assumed in classical 
theory). In turn, this implies that an electron must be able in 
travel long distances without being bumped by ions in the crystal 
lattice. The free-electron quantum theory, like the classical 
theory, is unable to account for the distinction between con- 
ductors and insulators. 



16.12 BAND THEORY OF CONDUCTORS, SEMICONDUCTORS, 
AND INSULATORS 

In the modem band theory of the electronic structure of solids, 
the effects of the lattice ions on the free electrons are considered 
to explain the occurrence of conductors, insulators, and semi- 
conductors. The moving electrons are pictured in terms of 






236 Looking In: Atomic and Nuclear Physics 

do Broglie waves of wavelength \ - h/mr. The influence of the 
lattice ions arises from the variation of potential from atom to 
atom in the crystal (Fig. Hi. 13). The passage of I he do Broglie 
waves is treated mathematically by methods similar to those 
used in investigating the passage of light waves through a similar 
lattice. 



Yinnrvv 



Fig. 16,13 Variations of poten- 
tial along a one-dimensional 
crystal lattice. 



It turns out that the graph of electron kinetic energy versus 
momentum, instead of having the parabolic shape (Fig. Hi. 14a) 
which it would have in a conductor where there was no variation 
of potential, jumps discontiuuously for particular values of 
de Broglie wavelengths (Fig. 16.14/*). Not all electron momenta 





(a) 



(b) 



Fig. 16.14 Energy vs. momentum: (a) Assuming no variation of 
potential between atoms; (b) assuming a variation of potential 
similar to that of Fig. 16.13. 

are possible. From this point of view, the effect of the ion lattice 
is to preclude certain values of electron momentum and hence to 
leave forbidden energy gaps at these momentum values. 

The properties of conductors, insulators, and semiconductors 
ran now be interpreted in terms of the conduction bands (Pig, 
Hi. 15). If the highest energy band containing electrons is full and 






Quantum Dynamics 237 

is appreciably separated from other bands (a), the materia! is an 
insulator. To produce a current in such a material, electrons have 
to be advanced across an energy gap large compared to thermal 
energy /,•'/'. In a conductor, however, the highest band containing 
electrons is not full (/*). Kven a small external electric field can 



Allowed (Empty) 



Allowed [Partly full 



Forbidden / £ w » « r 



Forbidden 



; (Allowed 
i) (Full) 



(a) 



(6) 



Allowed (empty) 

-Forbidden E 9 , p 2skT 
Allowed (full) 



(C) 



Fig. 16.15 Distribution of electrons in bands in (a) an 
insulator, (b) a conductor, and (c) a semiconductor. 

produce an unbalanced momentum distribution {a current) by 
promoting electrons to energy states of small excitation. Semi- 
conductors arc an intermediate case in which the highest occupied 
hand is full (c), but. the energy jump to the next band is compar- 
able to kT. Increase in temperature would be expected to lower 
the resistance of a semiconductor. 



17 



Radioactivity 



The new discoveries made in physics 
in the last few years, and the ideas 
and potentialities suggested by them, 
have had an effect upon the workers in 
this subject akin to that produced 
in literature by the Renaissance. 
J, J, Thomson, in an address on 
radioactivity, 1909 



Radioactivity has provided us with much of the knowledge we 
now have concerning the nucleus. Emission of a and ji particles 
by certain atoms suggested the idea that atoms are built of 
smaller units. Measurements of the scattering of a particles by 
atoms confirmed Rutherford's idea of the nuclear atom. The dis- 
covery of isotopes can be traced to the analysis of the chemical 
relationships among the various radioactive elements. The bom- 
bardment of atoms with energetic a particles from radioactive 
su Instances was found to cause disintegration of some atomic 
nuclei; this led in turn to the discovery of the neutron and to the 
present theory of the make-up of the nucleus. The transmuted 
atoms resulting from such bombardment are often radioactive. 

238 



Radioactivity 239 

The decay of these artificial radioactive nuclides is in accord with 
the laws found earlier in the study of natural radioactivity. 



17.1 TYPES OF RADIOACTIVITY 

In the theory of the nucleus there is no counterpart of the simple, 
easily visualized mechanical model employed in the Bohr theory 
of the atom. But the concept of energy levels, found so useful in 
studying the atom, is carried over to the description of the 
nucleus. Nuclear spectroscopy deals with the identification of 
these energy levels and is an important source of information 
about the nucleus, since radioactive changes can be measured 
with high precision. 

When the electronic structure of an atom acquires some extra 
energy, the atom almost always gets rid of this extra energy very 
quickly, returning to the ground state in roughly 10 - * sec. It does 
so by emitting one or more photons or an electron if there is 
enough extra energy. Many nuclei, however, can exist for long 
periods of time in an unstable state, that is, in a state from which 
the nucleus can and eventually will decay to a stable state. A 
nucleus may go to a state of lower energy by emitting an a 
particle (a radioactivity), an electron or positron (fi radio- 
activity), or a photon {7 radioactivity). 

Most "natural radioactivity" is found among the very heavy 
elements (A > 210), which tend to be unstable to a decay. 
These nuclei decay so slowly thai there are still some of them left 
from the time of formation of the elements. Radioactive isotopes 
not found in nature can lie prepared in nuclear reactions. 



17.2 STATISTICAL LAW OF RADIOACTIVE DECAY 

The activity of a radioactive sample is defined as the number of 
disintegrations per second. The activity decreases with time. 
Each radioactive isotope has its own characteristic rate of de- 
crease, figure 17.1 is the plot of the decay of a radioisotope which 
decreases in activity by 50 per cent every 4.0 hr. The form of t In- 
experimental decay curve suggests that the decay is a loga- 
rithmic process. This is verified by plotting the logarithm of 
activity versus time. A straight line results. 

We can derive an exponential law of decay for a sample con- 



240 Looking In: Atomic and Nuclear Physics 

tabling a large number of radioactive atoms. We assume that 
each undeeayed nucleus has a definite probability X of under- 
going decay in the next second and that this probability is inde- 
pendent of time and is independent of whatever other atoms are 
present. Then Iho number of decays in a time interval dt is equal 
to the number of undeeayed atoms present times the probability 









g 






< 






*m 






c 






3 






% 






^ 

fc 




K T 


"> 




\ t„=^L. 


t 




\ i " 0.693 


< 




1 i i i .i 



4 8 12 16 20 24 t, hr 

Fig, 17,1 Decrease in activity of a radioisotope 
with a 4.0-hr half-life. 



X (ll that each one of them will decay. Thus the change (decrease) 
in the number of undeeayed atoms is 



(IN = -\N(U 



(17.1) 



The decay constant X is the relative number dX/N of atoms 
which decay per second. The value of X depends only on which 
radioactive isotope we are considering, 

Uy separating variables in Kq. {17.1), we obtain a simple 
differential equation 



tlX 
N 



Xdt 



(17.2) 



whose solution is 
N = AV"*« 



(17.3) 



Radioactivity 241 

where N a is the number of undeeayed atoms in the sample when 
t = 0. (Note the mathematical similarity with the equation for 
the exponential ahsorption of a beam of radiation.) The activity 
of a sample, the number of decays per second, is given by 



Activity = -.:- = XAV -W 



or 



Activity = XN 



(17.4) 



(17.5. 



The activity depends on the number of atoms present and on their 
decay constant, X. 



17.3 HALF-LIFE 

The half-life T of a radioactive substance is the time interval in 
which the activity (and hence the number of undeeayed atoms) 
decreases by 50 per cent. For the activity of Fig. 17.1 this is 
4.0 hr. Itoiii the definition that t - T when X = IN^ Eq. (17.3) 
becomes 



hN a = AV" W 
which gives 

T = log, 2 = 0.«»3 



(17.6) 



(17.7) 



The average life '/'„, or life expectancy, of a radioactive nucleus 
may be calculated by Rimming the lives of all the nuclei and 
dividing by the total number of nuclei 



„, fi/ot dN I t - , A . . .. ., 1 

? « = / -57- = tt / 'Wr M dt = r 
Jo No No Jo X 



(17.8) 



The decay constant X is the reciprocal of the average life, in 
accord with the interpretation of X as the probability of decay of 
an atom per second. 



17.4 UNITS OF RADIOACTIVITY 

A unit of activity was historically defined as the amount of radon 
(gas) in equilibrium with one gram of radium. The National 



242 Looking In: Atomic and Nuclear Physics 

Research Council in 1948 extended this definition to define one 
curie as that quantity of any radioactive substance which gives 
3.70 X 10 10 disintegrations per second. Since the curie is a rela- 
tively large unit, the millicurie (I mc = 0.001 curie) and the 
microcurie (I ^c = 10 -s curie) are widely used. A counter near a 
radioactive source detects a certain fraction of the particles 
emitted; the counting rate is proportional to the activity of the 
source. 

The specific activity of a radioactive source is the rate at 
which 1 gin emits charged particles. 



17.5 GAMMA DECAY 



A nucleus in an excited statc(z*X i ) may go to a state of lower 
energy by emitting the difference in energy as a photon: 



zX* + hv C— M«v) 



(1 7.! I ) 



Now y decay does not cause a change in the atomic number or 
the mass number of the nucleus. The half-lives for y decay are 
seldom very long. 

Study of y radiation gives important information about the 
initial and final states of the nucleus undergoing a y transition. 
Like the spectra of atoms, the y spectra of nuclei are found to 
consist of sharp lines, showing that the nucleus has discrete 
energy levels. The observed energies of emitted photons give 
consistent results for the nuclear energy levels 

hv - Ei - E } (17.10) 

The electromagnetic-wave nature of y radiation is demon- 
strated experimentally by diffraction. This is feasible only for 
those 7 rays of relatively low energy because ruled gratings or 
crystals with effective spaeings about equal to very short y 
wavelengths are not available. 

The energies of high-energy 7 rays may be measured in 
several ways. When a 7 ray ejects a photoelectron from the inner 
shell of ati atom, 



hv = E k + I 



(17.11) 



where E k is the kinetic energy of the ejected electron and / is the 
binding energy of the shell from which it is removed. The ioniza- 



Radioactivity 243 

tion energies (/) are known. Hence the energy of the y-ray 
photons may be determined by measuring the energies of the 
photoelectrons. 

Positron-electron pairs (Chap. 19) can be created by 7 rays 
with hv > 2m c'. The photon energy is transformed thus: 



hv = 2m,e* + E k + + E k ~ + E k . tceoil 



(17.12. 



From conservation of momentum, the recoil velocity of the 
nearby nucleus should be small. Its energy can generally be 
neglected. Measurements of the momenta of the electron and 
positron in a magnetic field then give information from which the 
energy of the X ray can he found. 



17.6 ALPHA DECAY 

When an a particle is ejected from the nucleus, the original 

nucleus loses two protons and two neutrons. Its mass number 
decreases by four units while its atomic number Z decreases by 
two. « decay thus causes transmutation of the parent chemical 
element into a different chemical element 



Z X* -» «_,**-* + -.He' + Q (energy) 



-17.1:;. 



Now a decay occurs spontaneously, without any external forces, 
and it provides kinetic energy (#*,„) for the ejected n part ideas 
well as some kinetic energy {E k , d ) for the recoil ''daughter" nucleus. 
Hence a decay cannot occur unless the total rest mass decreases. 
The decrease in rest energy is equal to the kinetic energy released, 
called the disintegration energy Q: 



Q = E k ,,i + E k ,„ = (m„ - m lt - w Q )c ! 



(17.14) 



To predict whether a nucleus will undergo a decay, we may com- 
pare its rest mass with the sum of the masses of the product 
nuclei. Actually we can use the masses of atoms instead of those 
of the nuclei. The same number of electrons are associated with 
the initial and final nuclei, so the electron masses cancel in the 
calculation of Q. From Eq, (17.13), 



Q = (m* - mr — trine)*? 



(17.16) 



244 Looking In: Atomic and Nuclear Physics 

Exampk, Find the Q value for the disintegration t t>N'l u * —* tlic* 4- S8 Ce I4l >. 
Prom tables of isotope mosses: 



♦He 4 = 4.00387 

» a Ce l4 ° = 139.01977 

143.95364 



toX'!'*' - [43.95550 

Produeta - 143.95364 

m = 0.00192 

Q = mc* = 1.79 Mev 



Example. In a decay, what fraction of the disintegration energy appears 
as kinetic energy of (he a particle? 

Conservation of energy and conservation of momentum in a decay 
require 

Q = Et.4 + E kia = lmj>S + £fli a p a * 
m a v a = m&t 

From (he momentum equation, v d - (»ia/»ij)e„. Substituting this in the 
energy equal ion. ive have 






or 



Q 



" ^-'(S) 



+ 1 



AY~ = 



a 



1 + m„/m d 



(17.16) 



(17.17) 



If .4 is the mass number of the parent nucleus, then m a /m a =* 4/(A — 4) 
and 



**„ 



4-4 



(17.1 7«) 



Thus for large .4, the a particle gets most, but not quite all, of the dis- 
integration energy. 

An interesting feature of a decay called the tunnel effect may 
be illustrated by data for a particular ease. One can perform an 
experiment similar to the Kutlierford-Ceiger-Marsden scattering 
experiment (Chap. 15) using a thin foil of 94 U i!8 to scatter the 
7.68- Mev a particles from mIV 4 (also called Ra C). One finds 
that the Rutherford scattering law is obeyed. Kvtdently the 
a particles from IV 4 do not have sufficient energy to get over 
the Coulomb barrier; they are scattered away from the l"' :,s 
nucleus. This is suggested in Fig. 17.2, which shows the potential- 



Radioactivity 245 






/ u" 8 


t * 


n, 


t 


V, 


7.68 




1 Decay 


4.20 Me> 
1 


Mev 

» 


/ 






r 



Fig. 17.2 Coulomb borrier: scattering of a high- 
energy particle and tunneling of a low-energy 
particle. 







Fig. 17.3 Wave mechanical description of tunnel effect. 

energy curve of an a particle near a U m nucleus and a IV 4 
a particle being turned away by the potential barrier. Contrast 
this with the following fact: U 2,s itself is an a emitter, emitting 
a particles whose kinetic energy is only 4.20 Mev. We have a 
paradoxical situation : The lower-energy U S39 a particle can cross 
a barrier which the higher-energy I'o 414 a particles appear unable 



246 Looking In: Atomic and Nuclear Physics 



to cross. An explanation on the basis of classical physics is 
impossible. 

The wave nature of the a particle must be taken into account. 
When we use wave mechanics to describe an a particle in the 
nucleus, we find that a little of the wave function will "leak" 
through the barrier so that there is a small probability that the 
particle may be found outside (Fig. 17.8). According to wave 
mechanics, if the a particle has enough energy to be outside, 
then there is some probability that it will be found there. This 
probability is very small for U 2M and accounts, roughly, for the 
U 23 " 1 half-life of 4,f> billion years. The tunnel effect works in either 
direction, so some of the IV" a particles used in the scattering 
experiment must have penetrated the nucleus, but the fraction 
which succeeded was negligible. The probability of tunneling 
depends strongly on the height and width of the potential barrier. 

17.7 BETA DECAY 

The /3 particles emitted from a radioactive source are shown by 
deflection experiments to be high-energy electrons. There are 
good reasons to believe that, these electrons do not exist in the 
nucleus but are created by a rearrangement of the nucleus into a 
state of lower energy. Any excess of energy over thai required to 
provide one electron rest mass (m.c-) appeal's as kinetic energy of 
the emitted electron. 

An argument against the existence of electrons in a nucleus, 
prior to emission, makes use of the uncertainty principle. If an 
electron were confined in a region of dimensions no larger than 
about 2r = 1.4 X 10 _ " tn, the electron would have momenta as 
high as 

Ap = = 3.8 X 10 -' kg-m/sec = Um<c 

■ttt &,v 

and hence kinetic energy as high as 

E* = \/{Ap')*c s + m<c* - m.c' 1 = 14m,c s = 7.2 Mev 

It seems unlikely that there are attractive forces in a nucleus 
which are sufficiently strong to bind an electron having this much 
energy. 



Radioactivity 247 



Two different types of decay occur: ff~ decay, in which an 
electron is emitted from the nucleus, and (3 + decay, in which a 
positron is emitted. If the nucleus consists of neutrons and pro- 
tons only and if electric charge is conserved, then upon emission 
of an electron, a neutron must be converted to a proton, hZ = -f- 1. 
Similarly, positron emission involves the conversion of a proton 
to a neutron, AZ = — 1. 






(17.18) 
(17.19) 



For £~ decay to occur, the mass of the decaying nucleus must be 
greater than the mass of the product nucleus plus the mass of an 
electron. An atom which is heavier than the atom with Z one 
unit greater but with the same .1 will decay into that atom by 
0- emission. 

The condition for £+ decay is slightly more complicated. 



Q — nix — wiy — '2m r c 3 



(17.20) 



where m\ and m\- are the masses of the initial and final atoms, 
respectively, and hi. is the rest mass of an electron. An atom is 
ftf + unstable if it is more than two electron masses heavier than 
the atom with the same .1 and one less Z. 

There is still a third ($ decay process whose over-all result is 
the same as /J + decay. A nucleus may absorb one of its orbital 
electrons. This process is called A" capture since the elect Tons ill 
the nearest (re = 1) shell are most likely to be absorbed. The 
energy rule is the same as that for 0- decay: If the resulting atom 
is lighter than the original atom, it is unstable to K capture. 

The changes resulting from various nuclear processes are 
often represented in a proton-neutron diagram (Fig. 17.4) in 
which each nucleus is plotted in terms of the number (Z) of its 
protons versus the number (A — Z) of its neutrons. It is a result, 
of the processes we have just discussed that no two adjacent 
isobars (nuclei with same mass number) can both be stable. 
The heavier will #-decay into the lighter. 

The energies of electrons and positrons from decay have 
been determined with various types of /3-ray spectrometers. In 
principle, they measure the momentum of an electron by (hiding 
the curvature of its path in a known magnetic field. It is found 
that electrons in a given type of 8 decay may have any energy up 



248 Looking In: Atomic and Nuclear Physics 

















&T, 










~ 




Orig. 
nucleus 














K CQpt , 



















A-Z 



Rg. 17.4 A proton-neutron diagram. 



to the calculated energy release Q (Fig. 17.5), Here is a difficult; 
with the hypothesis that & decay consists of the emission of an 
electron (or positron) and the conversion of a neutron to a proton 
(or proton to a neutron). Tor the nuclear change is from one state 
of definite energy to another state of definite energy. Yet the 
electrons emitted carry varying amounts of energy, up to the 
maximum available. There is another difficulty. Consider the 
# decay of a nucleus containing an even number of nucleons. Its 
angular-momentum quantum number is an integer, since there is 
an even number of spin-i particles present. If a single electron is 




Fig. 17.5 A continuous spectrum. 



now created, there will be an odd number of spin-£ particles and 
the total angular-momentum quantum number will be half an 
odd integer. But a spontaneous change in angular momentum is 
not possible. 

To remove these difficulties, we assume that along with the 
electron, another particle, also of spin4, is created and emitted, 



Radioactivity 249 

but not observed! This particle is called the neutrino. Since it 
shares the disintegration energy Q with the electron, the con- 
tinuous energy distribution observed for the ff particles (Fig. 17.5) 
can be explained. The neutron is assumed to have zero rest mass, 
so the only change needed in our previous equations is to replace 
E k by E k + SfcneutrUo- The neutrino participates only in reac- 
tions. Since it has no rest mass, it travels with the speed of light. 
It is postulated to have spin ■$ and to obey Pauli's exclusion 
principle. The neutrino lias no electric charge, and it is difficult to 
detect! This remarkable particle has been assumed as necessary 
by physicists since about 1934. Its existence was first experi- 
mentally demonstrated in 1956, by detection of y rays produced 
in a planned sequence of events initiated by the neutrino. 



17.8 NATURAL RADIOACTIVE SERIES 

In experiments which followed the discovery of radioactivity, 
quite a number of substances were found to show activity. It was 
found that certain of these substances were associated with each 
other in series, the successive members being formed by the dis- 
integration of the preceding member, until a stable nucleus is 
reached. 

One can predict that there should exist four separate decay 
chains or radioactive series. A nucleus belongs to one of four 
classes, depending on whether its mass number A has the form 
4n, 4n -+- 1, 4» + 2, or 4n + 3, where n is an integer. Radioactive 
decay of a nucleus in one of these will result in the formation 
of daughter nuclei in the same class. This follows since there 
is no change in mass number in decay or in 7 decay, while 
in a decay, A.l = 4. The four radioactive series are represented 
in Fig. 17.6. Each bears the name of its longest-lived ele- 
ment. The neptunium series is not observed naturally because 
gaNp ! " (T = 2.2 X 10 s year) has almost completely decayed 
since tin- formation of the elements (about X 10' years tigo). 

The decay schemes of these four series end with stable isotopes 
of lead. A few radioactive isotopes which do not belong to the 
heavy-element chains are found in nature, Table 17.1. 

When the elements in a radioactive series are allowed to 
accumulate, a steady state will be reached (if the parent atom has 
a long half-life) in which the number Nx\i of atoms of one isotope 



250 Looking In: Atomic and Nuclear Physics 



N-A-Z 



Thorium series 
(A = 4n) 



? 



-z, 














>Th" J - 


140 








Ra 


Ht ( 


V A 








Lrv."* 
















Th 














14 


'Ho 














/• ' 








130 


Ph r; 


























Tl 




















Pb 


!0S 



























N = A-Z 

140 



Neptunium series 
(A-4H + I) Np J " 



130 











\ \ y\ 






















1 








R 


Li»«' 










Jf 














Ml 


r 








J"V 


/■A 










Pc s " 








Pb*"^. 


r" 










Tl™ 












1 













N=A-Z 



80 84 88 93 Z 

Uranium series 
(A = An + 2) yJM 



SO 84 88 92 Z 

Actinium series 
(A = 4n + 3) 




-6 














u ; 


j» 


















140 










Th iJ ' 




131 










Ac'V 


1 










* r itoJ 




Th' 37 














C 111 














Rr,*" 






130 








I s " 

Pc- 


15 








Pb ,u 


/ 












1 . 


X 


11 


























Hb 































80 84 88 92 Z 



80 84 88 92 Z 



Fig. 17.6 Decoy schemes of the four families of natural radioactivity. 

which decay per unit time is equal to the number A/oX* of atoms 
of the* next isotope which decay per unit time, or 

JV.Xi = N 2 \2 - iVaXa = ■ ■ ■ (equilibrium} (17.21) 

This equilibrium equation is often used to calculate X for an 






Radioactivity 251 






Table 17,1 Isolated natural radioisotopes 



Isotope 



Decay 



Half-life (years) 



isotope whose half-life is too large or too small to make a particle- 
counting experiment convenient. 

PROBLEMS 

1. Radium E has a half-life of 5.0 days. Radium E emits a 0- 
partiele to become radium P. (a) Which nucleus (E or F) has the greater 
positive charge? (b) Starting with 1.0 gm of radium E, how long would 
it take for -J gm to decay into radium F? 

2. a particles shot vertically upward arc deflected by the earth's 
magnetic field in which direction? 

3. Calculate the mass of Au l,s (7* = 2.7 days) in a source of 1.0 mc. 

Ans. 4.0!) X 10-" gm 

4. Five mg of IV '" {T - 140 days) are allowed to decay for 1.0 
year. What is the activity of the sample at Hie end of that time? 

Ahx. 1.35 X 10" disintegrations per second 

5. A sample of radioactive sodium (Xa- 1 . T = 14.8 hr) is assayed 
at 95 mc. It is administered to a patient 48 hr later. What is the activity 
at that time? Ans. 10 mc 

6. What is the volume of 1.0 mc of radon. M Hn m (T - 3.82 days), 
at 0°C and I atm pressure? Ans. fi.fi X 10~" m ! 

7. Suggest a method for using data on the uranium-decay series to 
estimate the age of the earth, Suggest B waj of using the radioactive 
isotope of carbon C u (T = 5,600 years) to substantiate the age of cot- 
ton fabrics found in an Egyptian tomb. 



iH 3 


r 


12.4 j Created continuously 
> by cosmic radiation 


.C" 


r 


5,590 J in atmosphere 


nK«« 


§,K 


1.2 X 10" 


^Rb" 


r 


6.2 X 10"> 


».lo ,li 


r 


6 X 10 14 


stLo"" 


r 


2 X 10" 


M Sm 147 


a 


1.5 X 10" 


7ito 176 


r 


2.4 X 10'° 


76 Re"< 


r 


4 X 10 12 









18 

Nuclear 
Reactions 



No man will ever comprehend the 
real secret of the difference between 
the ancient world and our present 
time, unless he has learned to see the 
difference which the late develop- 
ment of physical science has made 
between the thought of this day and 
the thought of that. T. H. Huxley 



A particle directed at a nucleus may undergo a collision (elastic 
scattering) which leaves the struck nucleus unaffected. A second 
possibility is that a nuclear reaction takes place producing sonic 
change in the struck nucleus. The incident particle may be 
absorbed into the struck nucleus. A rearrangement may occur in 
which the incident particle remains in the nucleus and another 
particle emerges. The incident particle may emerge but leave the 
nucleus in a different energy state. There are other possibilities. 
Nuclear reactions may be caused by individual nucleons, photons, 
deuterons, a particles, and heavier particles. 

252 



Nuclear Reactions 



253 



The first artificial nuclear transformation was achieved by 
Rutherford in 1919, in bombarding nitrogen with a particles 
from a natural radioactive source, Ha C Because of the impor- 
tance of neutrons in nuclear reactions, we shall depart from 
historic sequence to discuss first the discovery of the neutron by 
Chadwick in 1932. 

Among the achievements of nuclear studies are the production 
of scores of valuable isotopes, the discovery of the neutron and 
other particles, and the release of energy in the processes of 
nuclear lis.sion and fusion. 



18.1 DISCOVERY OF THE NEUTRON 

Bothe (1930) found that when a particles from polonium fell on 
a beryllium foil, a penetrating radiation was emitted. Irene and 
Frederic Joliot observed (1931) that the intensity of this radiation 
was apparently increased by passage through paraffin. They sug- 
gested that Bothe's radiation was y radiation which knocked out 
fast protons from paraffin and other hydrogen-rich substances. 

Chadwick (1932) applied the equations for the Compton 
effect to the head-on collision of the assumed y ray and proton 
(mass m) and showed that the maximum energy given to the 
proton by a photon (hv) would be 2hv/(2 +- mc*/hv). Experi- 
mentally the recoil protons from paraffin were found to have a 
maximum energy of 5.7 Mev, requiring thai the y ray from Be 
have energy hv ")."> Mev. Hut when aifcrogen was subotitvted 
for paraffin as a target, the i.'2-Mev recoil nitrogen ions which 
were observed required that the same y ray have an energy of 
90 Mev. Chadwick resolved this contradiction by suggesting that 
the "rays" from Be were actually neutrons, whose existence had 
been proposed by Rutherford in his mode! of the nuclear atom. 

The fact that atomic masses (beyond i\V) are roughly twice 
the atomic number suggests that the two types of particle 
neutron and proton which constitute a nucleus have approxi- 
mately equal mass. Chadwick confirmed this expectation by 
calculations made on the reaction 



5 B" + a IIe<-> ,»' + 7 N M + Q 



08.1) 



Three of the four masses were known. The energy of the incoming 
a particle (from Po) was known. The value of Q was determined 



254 Looking In: Atomic and Nuclear Physics 



Nuclear Reactions 



255 



from the observed increase in kinetic energy. The mass of tlic 
neutron was thus found to ho 1,00(57 amu. 



18.2 NUCLEAR FORCES; STABILITY OF NUCLEI 

The hypothesis thai atomic nuclei sire composed of neutrons and 
protons is now well established, and the term "nucleons" is used 
to refer to these nuclear particles collectively. The size of the 
nucleus is estimated by bombarding atoms with high-enerLiy 
electrons and counting Imw many of them score direct hits. The 
radius of a nucleus containing .1 nucleons is found to be 
approximately 



Iin, n = 1.2 X ID »,!' m 



(is. 2- 



An atom is stable because of the Coulomb force of attraction 
which binds the electrons to the nucleus. Within the nucleus, 
however, the Coulomb forces exerted by the protons are forces <>]' 
repulsion which tend to make the nucleus unstable. The emission 
of a particles from nuclei and nuclear fission (Chap. 20) are 
evidence of this. Somehow the repulsive Coulomb forces within a 
nucleus must be counterbalanced by strong attractive forces, 
different from electrical and gravitational forces. The nature of 
these nuclear forces is only partly understood. We shall discuss 
some of the facts which arc known about nuclear forces. 

An important, distinctive property of unclear forces is (heir 
short range. The nuclear force between two nucleons becomes 
negligible if they are separated by more than about 1,4 X 10 -16 m. 
In contrast, gravitational and electrical forces have no upper limit 
on the distances over which they may act. 

A second property of nucleus forces may lie deduced from a 
graph of the binding energy per nucleoli /•;,, .1 against the number 
of nucleons A (Fig. IS. I). Kxccpt for the lightest nuclei. E B .1 is 
approximately constant, about 8 Mev per micleon. Thus the 
total binding energy increases approximately in proportion to 
the number of nucleons in the nucleus: K„ a A. (The relation 
for a Coulomb force would be B« = A 2 .) This relation implies 
that a given nucleoli is bound not to every other nucleoli present, 
but only to its nearest neighbors. Then the addition of more 
nucleons increases the total binding energy only by an amount 



proportional to the number of nucleons added; K H /A does not 
change appreciably. 

Present evidence indicates that the nuclear force between two 
protons is the same as the force between two neutrons and that 
these may be equal to the force between neutron and proton. 

The last property of nuclear forces which we shall mention is 
pairing. The stable nuclei usually have even numbers of protons 



10 



■ 

* / 
■ — f- 



20 40 60 80 1 00 1 20 1 40 1 60 T 80 200 220 240 

A— 

Fig. 18.1 Binding energy per nudeon as a function of mass number A. 

and of neutrons (Table 18.1), Only the four light elements iH s , 
a Li°, sB 1 ", and 7 X U have odd numbers of both neutrons and 
protons, and for these elements the numbers of neutrons and 
protons are equal. 






Table 18,1 Evidence 


for pairing 


Neutron 
number (A — Z) 


Proton number Z 
Even Odd 


Even 
Odd 


160 52 

56 4 



256 Looking In: Atomic and Nuclear Physics 

When a plot of neutron number versus proton number is 
made for all nuclei (Fig. 18.2), one observes a gradual increase m 
the neutron/proton ratio with increasing Z. This is explained by 
the fact that the Coulomb (repulsion) force between protons 
increases more rapidly as the number of protons in the nucleus 
increases than does the effect of the nuclear force between protons. 
This difference in the behavior of the Coulomb pp force and the 
nuclear pp force accounts for the gradual decrease in E B /A from 

160 



140 



120 






Nuclear Reactions 



257 



i 

"* 100 



■a 
E 80 



p 

^ 60 



40 



20 



-t- « — 

/ • 

/ • 

/ s 

/ s 
/* 

£ . ■_ 



Fig. 18.2 Neutron -proton plot 
for stoble nuclei. 



20 40 60 80 100 
Proton number, X 



about 8.8 Me? for A near 50 to approximately 7.0 Mev for 

A = 240 (Fig, 18.1). 



18.3 NUCLEAR-REACTION EQUATIONS 

We shall consider some possible outcomes when a particle or 
nucleus ,r strikes a nucleus X resulting ha the emission of particle 
?/ and the obtaining of nucleus J": 



z + X— }■ + 



(18.3) 



The notation is often abbreviated as X(.c,y)Y, where the first 
symbol stands for the struck nucleus, the symbols in parentheses 



stand for the incoming and outgoing particles, respectively, and 
the symbol following the parentheses represents the residual 
nucleus. The reaction associated with Chadwick's discovery of 
the neutron, Eq. (18.1), may thus be abbreviated as Be 9 (a,n)C™. 
Before artificially accelerated particles became available, about 
1932, only 10 nuclear reactions were known, all of the (a,p) type. 
It seems probable that in the majority of artificially produced 
unclear reactions the first step is the formation of a compound 
nucleus. The projectile and the target nucleus coalesce. The com- 
pound nucleus is unstable, because of its excess energy. It emits 
one or, sometimes, more particles of high energy to regain stabil- 
ity. When Rutherford bombarded nitrogen with a particles 
emitted by Ra C (li)li)), he initiated the first nuclear transmuta- 
tion by artificial means. The equation describing it in terms of a 
compound nucleus is 

,He< + ,!?"-♦ [«F l *J-»«0" + ,11' (18.1) 

The same compound nucleus (but not in flic same energy state) 
could be produced by other reactions 

The breakup of the unstable compound nucleus usually depends 
only on its energy state, not directly upon the particle that pro- 
duced it. There are often several possibilities; for example, 

U*Zn 6i l -^ w&l« + T 

-+, Cu" + ,H J 

The 3 o*Zn Si may also eject other particles; ill 2 , iH*, sHe 1 or two 
n', but the probabilities of these reactions are low. Present 
nuclear theory does not permit prediction of the way a particular 
compound nucleus will break up. 

bike chemical-reaction equations, nuclear-reaction equations 
must be balanced. The total electric charge (the number of pro- 
tons) must be the same before and after the reaction. The total 
number of nucleons (neutrons and protons) must be the same, 
before and after the reaction. Together, these requirements mean 
that the number of neutrons must be the same before and after 
the reaction, likewise the number of protons. (There are two 



258 Looking In: Atomic and Nuclear Physics 

exceptions: If we regard ft* decay as a "reaction," then since there 
is no incoming particle, the number of neutrons changes by ± 1 
and the number of protons changes by + I. At extremely high 
energies, greater than 2 Hcv, it becomes possible to create micleou 
pairs. In such reactions, which we shall not discuss, the number 
of nucleoli* docs not remain constant.) 



18.4 THRESHOLD ENERGY 

In a nuclear reaction .r 4- .Y — ► 1" 4- >j, the net increase in kinetic 
energy is called the disintegration energy Q. This Q is the net 
decrease in rest mass, expressed as its equivalent energy: 



Q = H + m x ) - (m v + Tn. u )]c l 

Q = initial rest energy — final rest energy 



(18.5) 



Since Q is the amount of rest energy eon veiled into kinetic 
energy, Q is often called the energy release of the nuclear reaction. 
For an encounter which results in elastic scattering, Q = 0, If 
the Q value of a reaction is positive, the reaction is called exo- 
thermic. Such a reaction can occur for incident particles of any 
kinetic energy. If Q has a negative value, the reaction is called 
emlolhermic. 

Example. Calculate Hie Q value for the read ion T X 14 + n' -» 7 ,V l * 4- y 

,V - I i.odt.vji; 15.011)512 

n' = 1.00898(5 X' 6 = i;-).illMs7s 

16.016512 m = O.OlKiS-i amu 

Q = 931(0.01 KM) Mev = -H0.K Mev 

Conservation of momentum imposes a condition on induced 
nuclear reactions, as it does on all other collisions. This condition 
is particularly important for reactions with negative Q value. 
Prom energy considerations alone, one would think that ir the 
incident particle x approached the target nucleus (at rest) with a 
kinetic energy A'*.* = Q, then the reaction would occur. But then 
the momentum would not be conserved. The initial momentum 
is greater than zero, but the final kinetic energy, and thus the 
final momentum, would be zero. So, actually the incident particle 
must have enough kinetic energy B** so that the outgoing particles 
can have the same total momentum as the incident particle. The 



Nuclear Reactions 



259 



minimum value of A"*.., which makes the reaction possible is called 
the threshold energy. The minimum value of /;**. z which satisfies 
the equations for both conservation of energy and conservation of 
momentum is found to be 

Threshold - (£*.,) , ni n = (l + M Q (18.0) 

Example. Find the threshold energy for the reaction 
,|jm + )H i _* ,()!* + p „i (q = _3.4 S m,. v ) 

Threshold = ( 1 + — ) Q = ( 1 + ^§1)3.48 Mev = 3.72 Mev 

\ "».v/ \ 14.00// 



PROBLEMS 

1. State the number of protons and neutrons in each of the following 
nuclei: ,[,i* 6 I!e"», fi C 13 , 1S S 3B , and n Ui tm . 

2. The nuclear read ion 

,l.i + ,II»->2-.Hc* + Q 

liberates 22.4 Mev. Calculate the mas-, of JLa* in amu. (I)eutcron = 
2.014180 amu, a particle = 4.00:3873 amu.) 

3. Imagine that a free neutron gives off an electron and changes into 

a proton. Calculate the energy Q which is consumed or liberated in this 
process. What does your answer suggest about the stability of free 
neutrons? •Ins, Q = 6.79 Mev 

4. When neutrons :ire produced by bombarding deuterons with 
dcutcrons, the reaction is represented by 

,H*+ ,II i -. ! lle a + o" 1 + Q 

The neutrons produced in this reaction will have at least how uiurh 
energy? 

Am. 15 Mev plus the kinetic energy of the bombarding deuteron. 
.">. As the source of I he energy radiated by stars, it has been sug- 
gested that a series of nuclear reactions such as this carbon cycle occurs: 

C" + H' ..V + 7 Q =+l.»o Mev 

X la -» C 1 ' + e + + neutrino (K t $) , = 1.20 Mev 

C" + H' — X 14 4 7 Q ' = +7.58 Mev 

\u + H'^0" + 7 Q = +7.34 Mev 

O lb -* N" + e* + neutrino (A\ a),,,.,, = I .UN Mev 

K' B + II 1 -* C 12 + lb' Q = 4-4.98 Mev 

Write the equation which represents the net result of this whole cycle. 

Ans. 4H l — * He* 4- 2e v + 2 neutrinos 4- energy 



19 

Absorption 
of Radiation 



Science has a social value, and the 
man of science cannot wash his hands 
of his discoveries. It is his duty to see 
that they are used for the betterment 
of mankind, and not for its destruc- 
tion, Q Fournier 



To interpret experiments in nuclear physics and to apply the 
knowledge gained from them, it is necessary to know how the 
high-energy particles behave as they pass through mutter. For 
this discussion, high-energy particle means one whose kinetic 
energy is much greater than the ionization energy of the atoms or 
molecules of the material in which it is passing. We shall discuss 
the absorption of radiation chiefly in relation to the identification 
of particles, the measurement of radiation dost;, and the prob- 
lems of human health. 

19.1 TYPES OF RADIATION 

Jn the behavior of a high-energy particle the most important fact 
is whether or not it carries an electric charge. A particle which 

260 



Absorption of Radiation 261 

carries an electric charge (as do the electron, positron, proton, 
deuteron, and a particle) will exert a force on each electron near 
which it passes. A charged particle collides with many electrons 
in traveling even a short distance in matter. In many of those 
collisions, the struck electron is knocked out of its atom. The 
incident charged particle loses its kinetic energy as it leaves 
behind a trail of ion pairs (ejected electron and ionized atom). A 
stream of charged particles is referred to as an ionizing radiation. 
Photons and neutrons which carry no charge do not necessarily 
collide with every electron near their paths. Streams of uncharged 
particles are called nonionizing radiation. 

19.2 DETECTORS 

Ionizing particles are easy to detect electrically. In an ionization 
chamber, a metal cylinder C has a wire II' insulated from the 
cylinder along the axis. The tube is filled with gas at low pressure, 
and a potential slightly less than that reiumed for a discharge is 
maintained between cylinder and wire. A thin window allows 
particles, say, a particles, to enter the chamber. Kach particle 
ionizes the gas, producing a rush of charge and a fall of potential 
at P which actuates a counter circuit. Thus one can count the 
number of a particles. The behavior of the ion pairs created can 
be studied by plotting a curve of the size of the current pulse 
versus the voltage applied to the tube. The ionization chamber 





Bottery ; 



1 



Capacitor 



# 



To amplifier 
and counter 



resistor 



Fig. 19,1 Ionization chamber particle counter. 



(Fig. 19.1), the proportional counter, and the Geiger-.Muller 
counter are ionization instruments designed to operate on dif- 
ferent regions of the curve. 

A scintillation counter makes use of one of several substances 



262 Looking In: Atomic and Nuclear Physics 

which, when struck by a single particle, convert some of the 
energy received in the collision into visible tight. About. ]«)()•( 
investigators <rf radioactivity watched and counted the flashes of 
light which individual a particles produced in zinc sulfide. Since 
l!)-l I a scintillator or phosphor such as a clear crystal of naph- 



Al foil reflectors 

Photo cathode 
semi transparent 




First dynode 



Tenth dynode JJ "£ -".-.j— - "- Col lector grid 



Output 
Fig. 19.2 Scintillation counter. 

thalene has been used in conjunction with a pilot onmlfiplier, for 
automatic counting. A particle or a -y-ray photon entering the 
phosphor causes a flash of light which is reflected by the aluminum 
foil onto the photocallmde. Klcctrons are emitted from it, and 
these are subsequently multiplied to produce a relatively large 
pulse at the output of the tube. 

A cloud chamber, invented by C. T. It. Wilson in 18117, per- 
mits us to see the path of a particle through a gas. It consists of 
an enclosure filled with air and some vapor at a temperature 
just above, the condensing temperature. The chamber is designed 
so that its volume may be suddenly increased. This expansion 



Absorption of Radiation 263 

depresses the temperature of the vapor below its "dew point." 
Some of the vapor will now condense. A vapor condenses prefer- 
entially on charged particles, as nuclei for droplets, if there are 
any present. So, if the gas has been traversed by a particle which 
ionized molecules along its path, the vapor will condense on these 
ions and the path of the particle will be visible as a trail of liquid 
droplets. 

Photographic plates were used by Bccquerel in his discovery 
of radioactivity (1886). Recently the manufacture of special 
emulsions for nuclear research has revived the use of this type of 
detector. Nuclear emulsions contain about 10 times the concen- 
tration of silver halide as do ordinary photographic emulsions, 
and are much thicker. Xuclear emulsions can be made sensitive 
to slow neutrons by incorporating small amounts (I per cent) of 
lithium or boron, which undergo an (n,a) reaction. Emulsions 
may be "loaded" with other elements (such as uranium) to study 
specific reactions. In film badges, the general darkening of the 
photographic emulsion, on development, measures cumulative 
exposure to radiation. In autoradiography, the distribution of 
radioactive material in a tissue or mineral section is determined 
by placing the specimen in contact with a photographic plate, in 
the dark, and developing the resulting pattern. 

The bubble chamber, invented by D. A. Glaser in 1952, takes 
advantage of the instability of superheated liquids for bubble 
formation, much as the Wilson cloud chamber uses the instability 
of supercooled vapors for droplet formation. The cloud chamber 
and the bubble chamber have similar general characteristics as 
particle detectors. The resetting time is longer than lor counters. 
The advantages of the bubble chamber lie in the high density 
(greater absorption) of its sensitive material and its ability to 
recycle in a few seconds. Bubble chambers filled with liquid 
hydrogen offer simplicity in interpreting collisions with protons, 
without contaminating elements. 



19.3 DETECTION OF NEUTRONS 

A neutron is attracted to other near nucleous by the nuclear 
force, but it is neither attracted nor repelled by an electric charge. 
Since a neutron and an electron exert no forces on each other, 
they do not collide. (We can neglect for practical reasons the 



264 Looking In: Atomic and Nuclear Physics 

extremely small gravitational force between an electron and a 
neutron and also a small electromagnetic force associated with 
the magnetic moments of the two particles.) Since nuclei occupy 
only a small fraction of the volume of matter, neutrons are pene- 
trating radiation, traveling relatively large distances between 
collisions. When a collision does take place, either the neutron 
is scattered or a nuclear reaction occurs. 

Since neutrons do not betray their presence directly in de- 
tectors (Sec. 19.2), they must he detected by the ionization which 
results from some nuclear reaction of scattering. For slow neu- 
trons (having kinetic energy less than I ev) it is convenient to 
use the reaction 

JB»-f oNi-» Ji»-|- ,He l 

If a counter tube is filled with a gas containing boron, BK 3 , or if 
the wall is coated with boron, then some neutrons will he captured 
to give fast a particles, which will cause ionizations in the gas 
and give counts. 

Another method used to detect slow neutrons makes use of 
the reaction 

on 1 + ^In' 16 -* win" 6 + y 

The radioactivity of an indium foil after exposure to a neutron 
beam is a measure of the number of neutrons which passed 
through the foil. The (n,y) cross section, or probability of cap- 
ture, is sharply higher for neutrons of l.4(S-ev energy. Thus this 
detector favors or picks out those neutrons. 

The detection of fast neutrons, and the initiation of certain 
important reactions, often requires first that the neutrons be 
slowed down. This is accomplished by arranging for the neutrons 
to pass into a moderator— a material such as graphite or D,,0 in 
which the probability (cross section) for scattering is much larger 
than that for a nuclear reaction. The neutrons then bounce 
around among the nuclei until both reach an average energy of 
!i/,-r, where k is the Boltzmaun constant. 

Bxampk. Find the energy of a "thermal neutron" in n moderator at 
22 C 

B k = |(1.3H X 10-" joulc/K°)(295°K) • 6.11 X 10~" joule 
= 0,0382 ev 



Absorption of Radiation 265 



19.4 ABSORPTION OF PHOTONS 

Photons can interact directly with the electrons of the material 
through which they pass. But for high-energy photons, the cross 
section (probability) of such interact ions is so small that the 
photons constitute an extremely penetrating radiation. The 
energy of photons can lie dissipated in three different kinds of 
collision. 

In the photoelectric effect a photon is absorbed by an atom; its 
energy is used to eject an electron and to impart kinetic energy 
to the electron. The cross section for the photoelectric effect 
increases rapidly with increasing atomic number (Z) and de- 
creases rapidly with increasing energy (hr) of 6he photon. 

In pair production, the energy of the photon is converted into 
a positron and an electron and their kinetic energies. The cross 
section for pair production increases rapidly with increasing Z 
of the absorber and with increasing energy of the photon, above 
the threshold value of 1 Mev (= 2m c 2 ). 

In the Campion effect, photons are in effect scattered, not 
absorbed. A photon is still in play after the collision. The cross 
section is a slowly varying function of (hv) and Z. 

The detection of photons is relatively simple; for any type of 
collision described above gives a fast electron: a photoelectron, 
a Compton electron, or an electron-positron pair. The electrons 
are ionizing particles and may be counted directly. 

The variation of photon "absorption" by each of these proc- 
esses is represented in Fig. U)M, where for each process, an absorp- 
tion coefficient a is defined as the product of the cross section a of 
the reaction and the number n of atoms per unit, volume, a — rur. 
If the Compton effect were strictly an absorption, a total absorp- 
tion coefficient a, could be defined for photon absorption 



Ctt — OfphutM "T" G^air I ^Complin 

ami the attenuation of a beam of x rays or 7 rays could be repre- 
sented by the exponents! equation 



While this relation has practical usefulness, it must be applied 
with care, since eeoinpton does not relate to a true absorption. 



266 Looking In: Atomic and Nuclear Physics 
a, cm" ' 





|. 


1.4 


i I l a niol 




\ ^*m 




^ 1 J* 


1.2 


■ ^ 1 1 /? 




\ ' \ A 


1.0 
0.8 


\ 1 V Jya. ra „ 




» x \ A 


0.6 


- \%~4 




0.4 


V s / 


0,2 








v^ -»-»_ 







0.5 5 50 

Photon energy 



500 Mev 



Fig. 19,3 Variation of photon absorption coefficient, a, in 
load, with photon energy. 



19.5 RANGES OF HEAVY CHARGED PARTICLES 

Charged particles heavier than electrons experience frequent, 
collisions with electrons in passing through matter. The heavier 
particle cannot lie appreciably deflected, and il can lose only a 
small fraction of its energy in collision with an electron. Vet the 
collisions are so frequent that charged particles are slowed down 
to thermal energies in very short distances. Charged particles are 
not a penetrating radiation. A proton with 10 Mev of kinetic 




Distance traveled, S 

Fig. 19.4 Kinetic energy vs. distance troveled for a charged 
particle. 



Absorption of Radiation 267 

energy travels only 0.0 mm in aluminum; a 10- Mev a particle 
travels only 0.00(> mm in aluminum. 

The decrease in the kinetic energy of a charged parlislc with 
distance traveled is indicated schematically in Fig. 19.4 as 
occurring in many small steps. The distance traveled before the 
kinetic energy is all lost is called the range of the particle. Range 
depends on the particle, its initial energy, and the absorbing 
material. When the kinetic energy of the charged particle has 
been reduced to a small value (about 100 ev for a proton), it 
becomes increasingly probable that the ion will capture an elec- 
tron and end as a neutral atom. 



19.6 ABSORPTION OF ELECTRONS AND POSITRONS 

The path of an electron or positron is longer than that of a heavy 
charged particle of the same energy, but it is a path full of bends 
because of scattering. Electrons, like other charged particles, lose 
their energy in a very small region of space; they do not constitute 
a penetrating radiation. 



19.7 RADIATION DOSE 

The dose of any kind of radiation received by an object is the 
amount of energy that the object absorbs from the radiation. 
One might try to use a calorimeter to measure the energy ab- 
sorbed by a specimen in terms of the resulting rise in its tempera- 
ture. It turns out that- even a lethal dose of radiation produces an 
undetectable rise in the temperature of a biological specimen. 

Radiation produces many specific effects on physical, chem- 
ical, and biological systems. Many of these effects seem closely 
related to the ability of the ionization caused by the radiation to 
promote particular chemical reactions. Hence methods have been 
devised to specify dose in terms or ionization. 

A beam of x rays or y rays is said to give a dose of one roentgen 
(1 r)* if it will cause 2.08:{ X 10" J ionizations in 1 cm 3 of dry air at 

"The National Bureau of Standards Handbook H47 gives the defi- 
nition: "The roentgen shall be the quantity of x or y radiation such that 
i he associated corpuscular emission per 0,001293 gin of air produces, in 
air, ions carrying 1 esu (if quantity of either sign," The figure 0.001293 



268 Looking In: Atomic and Nuclear Physics 



Absorption of Radiation 269 



0°C and 1 atm. An ionization chamber is used to measure the 
dose from the radiation. The radiation passes through the air 
between the plates, and the ionization occurring in the air is 
collected. The chamber and its electrometer can be calibrated to 
read directly in roentgens. A widely accepted human tolerance 
dose rate is 0.3 r per week. The dose from cosmic rays at the 
surface of the earth is about 2 per cent of this tolerance dose. 

The roentgen was defined for photons. To extend the unit to 
permit measurement of radiation dose from other particles, and 
in living tissue, the roentgen equivalent physical (rep) is desig- 
nated as the radiation which produces the same energy as one 
roentgen of x- or 7-radiation. This amounts to 97 ergs per gram 
of tissue. This value is based on the observation that for any 
particle and any gas the average energy lost by a fast charged 
particle per ion pair formed is about 33.5 ev, A third unit for 
radiation dose is the red: the radiation which produces LOO ergs 
per gram of tissue. 

19.8 BIOLOGICAL EFFECTS OF RADIATION 

Living tissue is damaged 1 >.V exposure to high-energy radiation. 
The danger is insidious, for the observed biological effects may 
be delayed for periods ranging from a few days to years, depend- 
ing upon the type of radiation and the dose received. Among the 
effects of overexposure to radiation are a decrease in the number 
of white blood cells, loss of hair, sterility, cancer, cataracts 
(chiefly from neutrons), and destruction of bones. In addition to 
the damage to the person receiving the radiation, there may be 
genetic effects extending through many generations of offspring. 
Penetrating radiations are effective in producing mutations or 
changes in heredity. 

X rays, y rays, and particles from supervoltage accelerators 
penetrate tissue readily and constitute externa! radiation hazards. 
In general, « and /S particles have low penetrating power, and 

Kin is the muss of I cm 3 of dry air at 0°C and 1 atm. Since 3 X 10 s esu 
of charge = 1 coul, I r produces 

1 

3 X 10» statcoul/coul 1.6 X 10-" coul/ion 

= 2.083 X 10» ion pairs/cm' 









damage from external sources will be confined to a thin layer of 
tissue. But a and emitters become internal hazards when intro- 
duced into the body in foods or otherwise. 

The various kinds of radiation damage seem to he statistical 
in nature, with no threshold or "safe" minimum exposure below 
which no injury occurs. Hence it seems prudent, to avoid all 
unnecessary radiation exposure. Since some exposure may be 
necessary for some people, responsible agencies have suggested 
tolerances, such as a whole-body exposure of 0.3 r per week when 
continued over a long time. I'or hands and feet the tolerance may 
be 1.0 r/ week. A single exposure of 25 1- in an accident can prob- 
ably he accepted. A whole-body exposure of about 500 r would 
probably be fatal, statistically, to 50 per cent of persons so 
exposed. 



19.9 ATMOSPHERIC CONTAMINATION FROM NUCLEAR 
WEAPONS TESTS 

The probable effects on the health of the world population of 
atmospheric contamination arising from nuclear weapons tests 
cannol be assessed reliably from data known at present. Vet on 
the basis of incomplete information and conflicting interests, 
political decisions about nuclear detonations must be made which 
vitally affect our national defense and the freedom and health of 
generations to come. 

If one examines, in addition to research reports, some 10 
official statements made since 1 !>.">(> by the Congressional Joint 
Committee on Atomic Energy, the United Nations Scientific 
Committee on the Effects of Atomic Radiation, The National 
Research Council, and the (British) Medical Research Council, 
one finds that these responsible bodies are in agreement on the 
following points: 

1. Radiation exposure of the world population from fallout 
(including Si'"") as a result of tests through mid-1963 is small 
compared to natural background radiation and other man- 
made radiation (such as diagnostic x rays). 

2. Any amount of radiation, however small, may carry a small 
but finite risk of increasing the genetic mutation rate of the 
population. 



270 Looking In: Atomic and Nuclear Physics 

3. Tt is unknown whether or not there exists! a threshold radiation 
dose for the production of somatic effects, including leukemia, 
bone cancer, and general life shortening. 

4. Calculations of biospheric contamination in the event of con- 
tinued testing of nuclear weapons are intelligent guesses at 
best, since conclusions depend on the many assumptions that 
must be made. 

5. Continued testing of nuclear weapons will increase biospheric 
contamination and consequent risk to the world population. 
Accelerated testing as more nations become nuclear powers, 
and (he touching off of nuclear war, could result in a serious 
radiation hazard to world health. 



19.10 DISPOSAL OF NUCLEAR WASTES 

Nuclear power ranuol be developed by present techniques with- 
out also producing radioactive waste materials which are harmful 
to man. The safe disposal of such radioactive wastes is far more 
difficult than that of ordinary industrial wastes. More than 
ti;5 million gal of highly radioactive nuclear wastes are now con- 
fined in million-gallon underground tanks because they are too 
"hot" to dump. Although the concrete and steel tanks are ex- 
pected to last several decades, their contents will still be too 
radioactive to dump when the (auks have deteriorated! 

There has been increasing local public protest against the 
dumping of nuclear wastes into the oceans, relatively close to the 
shore ; particularly by citizens ul Cape Cod, Texas, and Mexico. 
It has also been pointed out that it may even be dangerous to 
dump nuclear wastes in remote and deep trenches of the oceans 
because (I) experiments increasingly indicate thai there is con- 
siderable circulation of ocean waters and (2) marine organisms 
tend to build up small and nearly harmless radioactive levels in 
sea water to potentially dangerous levels in the food supply. 

At the present time there are four general sources of radiation 
which can harm the present and future generations. In order of 
intensity, these are (1) medical and dental x rays, (2) radioactive 
sources naturally present in the earth, (3) radioactive fallout from 
nuclear testing, and (4) waste products from nuclear reactors. 
Within a decade or two, the latter two sources of radiation 
exposure may become the most important. 



Absorption of Radiation 271 



SUGGESTED READING 

Articles in the Bulletin of the Atomic Scientists. 

The Milk We Drink, Consumer Reports, March, 1959. 

Fallout, in Our Milk, Consumer Reports, February, 1960. 

The Huge and Kver-iiiereushig Problem of Radioactive Wastes, Con- 
sumer Reports. February, I !)(!(). 

Fallout 1963 . . . an interim report, Consumer Reports, September, 
19(iH. 

I^utgham, Wright, and B. 0. Anderson; "Biospheric Contamination 
from Nuclear Weapons Tests through 1968," Los Alamos Scientific 
Laboratory, University of California, I.os Alamos, X.Mex. 100 pp. 
Contain.* bibliography of 7 I ilems. 



I often say that when you can measure what you are speaking about and 
express it in numbers, you know something about it; but when you cannot 
express It in numbers, your knowledge is of a meagre and unsatisfactory 
kind; it may be the beginning of knowledge, but you have scarcely, in your 
thoughts, advanced to the stage of science, whatever the matter may be. 

Lord Kelvin 

Life would be stunted and narrow if we could feel no significance in the 
world around us beyond that which can be weighed and measured with the 
tools of the physicist or described by the metrical symbols of the mathe- 
matician. Sir Arthur Eddington 



Accurate and minute measurement seems to the non -scientific imagination 
a less lofty and dignified work than looking for something new. But nearly 
all the grandest discoveries of science have been but the rewards of accurate 
measurement and patient long-continued labor in the minute sifting of 
numerical results. Lord Kelvin 



It does not take an idea so long to become "classical" in physics as it does 
in the arts. K. K. Darrow 



20 

Unconventional 
Energy Sources 



. . . the discovery with which we are 
dealing involves forces of a nature too 
dangerous to fit into any of our usual 
concepts. Congressional Record, 
T87S, commenting on the gasoline 
engine 



A physicist, like other persons, often finds living more purposeful 
and satisfying when he haw both short- and long-range goals. 
Some physicists seek to relate their goals to some of civilization's 
long-range problems: food production, world peace, education, 
and the exploitation of new sources of energy. It would seem that 
physics could contribute most directly in finding new sources of 
energy to supplant depleted reserves of coal and oil and to meet 
the ever increasing demand for power for industry, transporta- 
tion, and the home. Since we never create energy, it might be 
more precise to speak of a search for new and practical energy- 
conversion devices. 

Some possible sources of energy are so speculative that they 
are referred to as esoteric sources. The term "unconventional" is 

272 



Unconventional Energy Sources 273 

reserved for those untapped sources ahout which enough is 
understood today so that one may reasonably predict that 
engineering refinements will soon make of them practical energy 
sources, important in our economy. Nuclear reactors, thermo- 
electric, thermionic, ami magnetobydrodynamic generators, solar 
cells, and fuel cells give promise of becoming increasingly impor- 
tant practical sources of energy. 



20.1 NUCLEAR FISSION 

When, in 1042, the book "Applied Nuclear Physics" (K. Pollard 
and W. L, Davidson) was published, its title sounded visionary. 
Since then we have witnessed important and varied applications 
of nuclear physics. The nuclear reactor has heen developed into a 
practical source of electric power. (A reactor may become the 
ultimate source of power for space travel.) With particle acceler- 
ators and nuclear reactors, a host of new isotopes have been 
created. These have been important in further fundamental 
studies. They have also found diverse practical applications. 

In 1934, Fermi and his collaborators attempted to produce 
elements beyond the normal limit at uranium. In bombardment 
of the lighter elements by slow neutrons, the element after the 
capture is usually transformed by electron emission into the ele- 
ment of next higher atomic number. Therefore, one might expect 
that a similar bombardment of uranium (Z = 92) would produce 
a new element (93). This reaction has been produced with 
neptunium (93) as the resulting product. Neptunium also dis- 
integrates by emitting a (i particle to produce plutonium (94). 
Plutonium is a rather stable clement having a half-life of 24,400 
years. From 1944 to 1950, four other new elements were produced 
in the cyclotron: americium (95), curium (96), berkclium (97), 
and californium (98). More recently elements einsteinium (99), 
fermium (100), mcndelevium (101), and nobelium (102) have 
been reported. 

In 1939, Halm and Strassmann found one of the products of 
neutron bombardment of uranium to be a radioactive barium 
sijBa 139 . There must then be another fragment such as 36 Kr associ- 
ated the barium fragment to make the charges equal. Ncir 
separated the isotopes of uranium in a mass spectrograph and 
found that «U s,b is the one that undergoes the splitting process 



274 Looking In: Atomic and Nuclear Physics 

called fission. Fission is a new type of radioactive process, the 
first that produced particles more massive than a particles. 

In the process of fission of uranium there is a decrease in total 
mass, and therefore there is a corresponding gain in energy. Such 
a reaction then is a possible source of energy. This energy is con- 
trollable since the process can be started at will and its rate can 
be governed. 

Among the products of fission one finds one to three neutrons. 
These neutrons are faster than the ones used to start the fission, 
but if they strike uranium nuclei, they can cause fission. Since 
the fission produces the starting particles and releases energy, 
the reaction can perpetuate itself, provided there is enough 
uranium present so that the neutrons produced will hit other 
uranium nuclei. Thus a chain reaction can be set up. The smallest 
amount of material in which a chain reaction (constant neutron 
flux) can be set up is called the critical mass. 

20.2 NUCLEAR REACTOR 

A nuclear reactor is a device for utilizing a chain reaction for 
any of several purposes : to produce power, to supply neutrons, 
to induce nuclear reactions, to prepare isotopes, or to make 
fissionable material from certain "fertile" materials. Typical 
components of a reactor are: the fissionable fuel (LI or Pu), the 
moderator (graphite or D a O to slow down the fission-producing 
neutrons), the control rods (usually Cd strips, whose insertion 
captures neutrons and slows the fission rate), and the coolant 
(water, air, hydrogen, or liquid metal, such as \a). 

In power reactors, the coolant, through a heat exchanger, 
may furnish steam to operate a conventional turbine and elec- 
trical generator. Breeder reactors make new nuclear fuel from 
fertile substances which cannot themselves sustain a chain reac- 
tion but which can be converted into fissionable material. One 
possible breeding reaction is 

iNP^^MPO** (20-1) 



T -, 
* t^l. 



2'A ruin 



'2:.i duy» 



20.3 FUSION 









Nuclear energy can also lie released by fusion of small nuclei into 
larger nuclei if in this process there is a decrease in mass. In such 



Unconventional Energy Sources 275 

a process the two positively charged nuclei must come into con- 
tact even though there are strong electrical forces of repulsion. 
This requires thai I lie particles he moving with high speeds. With 
artificial accelerating apparatus, a few nuclei are given very high 
speeds. Only occasionally will such a particle strike another 
nucleus before it has lost too much of its energy to make contact. 
Thus the process is extremely inefficient, and more energy must 
be supplied to initiate the fusion process than is realized from the 
reaction. 

The necessary condition for a controlled nuclear-fusion process 
is the attainment of high particle energies for a time interval long 
enough to bring about kinetic equilibrium. Knergy must be sup- 
plied initially to attain temperatures about 2 X 10 7 °K (at which 
thermal fusion occurs in stars). At the same time reactants must 
be confined. Ordinary walls will not suffice, for they would 
vaporize under bombardment of high-energy particles, and these 
would be quickly cooled below their fusion temperature. These 
problems of heating and confinement must be solved in any con- 
trolled-fusion reactor. 

The choice of fuel for a eon t rolled-fusion reactor is made on 
the basis of availability and the probability of attaining with it 
the necessary high temperature. One would prefer elements of 
low atomic number because of the low Coulomb barrier to be 
overcome in the fusion reaction. Possible fusion reactions are 
shown in Fig. 20.1. 

Initial heating first strips the electrons from the atoms to pro- 
duce a "fourth state of matter," a fully ionized gas, or plasma. 
Further heating of the plasma is done by adding electric energy, 
in part by using the resistance of the plasma to produce familiar 
Ohmic (or Joule) heating. 

Suitably designed magnetic fields provide a sort of magnetic 
bottle to confine the ions at I0*°K. In the pinch effect, a cylin- 
drical current (10° amp) contracts because of electro magnet it- 
forces (parallel currents attract each other). The plasma inside is 
thus compressed, producing very high temperatures. The simple 
pinch is unstable, but with suitable stabilizing fields thermo- 
nuclear temperatures have been attained for confinement time 
of about 0.001 sec. Thus far, however, the power required for 
these devices has exceeded the useful power gained from the 
fusion process. 



THE FOUR STATES OF MATTER 



1-Solid 



2- Liquid 



3- Gas 



4-Plosrao 



tlili 










First .hres itafei of roaMer vory with arrongenien. 
and movement of maleculei, ihe ima'lleir parr.c1.ev 
C K« roc t •>■ lii ; c of a rrva i«f io I . tn to I id , mo lee u 1 ei 
or* cfoYe-packec 1 ond trill ■ In liquid they tnovt 



about within limiti. In a got, moleculei ore man 
(coflered Qnd movv Foiier, Fooith lfoie^ ploinna, it 
wholly '"ionized" 901. Molecule* break into aTorm, 
alonm into poiilive ion* and rega-ive elecrron>>. 



• Proton 



O Neutron 



-THE FUSION REACTION- 
Deuteriurn Fusion 

He3 + 




P 



Energy 



En ergy 



3,25 
Mev 

■ 4 Mev 



Pi^rC^ 



Deuterium -tritium fusion 



He* 



^^5 M^ 



fusion con take piece within a plasma, Fuiian ii 
combination of nuclei (atom* minus electrons} of 
certain lighl element!. Man of the Fuiion product! 
rl let* than that of orioinal nuclei; the difference 
h radiated os energy, mojlly heat. The fcjn"« voit 



1 HEATING THE PLASMA 


■;'-..: - Direction of current-. 




. •-■■.... 





To get controlled ihermonucleai reaction in- 
stead of explosion, small quonliriei of plasma 
must be contained and heated. Process begins 
with passage 0! a current through the plasma 
inside a Straight nr doughnut -shaped tube. 



energy is from hydrogen fusion . On earth 
most likely such reaction involves Twtopes 
(voiiants) of hydrogen — deuterium and 
tritium. In o plasma heated to millions of 
degrees, they may fuse, as in the H-bomb. 

THE PINCH EFFECT 



-PROBLEM OF INSTABILITY- 




mm 



A tuirenl eieaies o magnetic field around itself. 
This Field exerts pressure on plasma, "pinching," 
it toward center,, compressing St, making it hotter 
and preventing plasma pari ides from touching 
walls af tube . But this is theoretical behavior. 



-CONTROLLING INSTABILITY- 




In practice pinched column of ploimo develops 
"kinky," Pinch wandert llighHy; distortion af 
■he magneric Held create* new forcer and diirorn 
column further. Pinch eilher rouchei walk of r^re 
rub* and loiei energy (A) or ii broken or or (B) . 



Powige of new current (A) around rube create! a 
linear meaner ic field in column/ giving il "back- 
bane/' Currenli induced in wo 111 of lube (B) help 
uraighren column. Pinch can then be mointoined 
longer. 



Fig. 20.1 Principles of a thermonuclear reaction, f Copyright by Trie New/ York 
Times. Reproduced with permission.) 

276 



Unconventional Energy Sources 277 



20.4 THERMOELECTRIC CONVERSION 

The direct conversion of heat to electricity on a commercial scale 
is a prospect that has fascinated scientists and engineers for 
decades. In 1821, Thomas Sccbeck noted thai heal -applied to 
one junction of a circuit containing dissimilar metals would cause 
a small electric current in the connected circuit. The physical 
median ism can be understood, qualitatively, in terms of the free- 
electron picture of conduction. Kach metal contains some free 
electrons. These electrons can be made to move by an electric 
field or by a thermal field. If heat is applied at one end of the 
conductor, the electrons will rearrange to become somewhat more 
sparse in the warmer regions of the .specimen and more dense in 
the colder regions. This leads to an electrical gradient. To take 
advantage of it, the circuit is closed through a dissimilar metal 
(Fig. 20.2). Then, as long as the temperature difference is main- 
tained, the difference in electrical gradient in the two conductors 
will cause an electron flow, here clockwise. 

The efficiency of conversion, using the best metal combina- 
tions, was only 1 to 3 per cent. Thus, until recently the only 
practical application of Seebeck's effect was in thermocouples to 
measure temperatures. Recent discoveries in the field of semi- 
conductors have led to substantial improvement in thermoelectric 
conversion efficiency and foreshadow practical thermoelectric 
generators of power. One arm of the thermocouple may be made 
of an «-type semiconductor, in which the voltage difference is 
established by the flow of negatively charged electrons. The other 
arm may be a p-type semiconductor in which the voltage differ- 
ence occurs by the flow of positively charged voids (holes) 
vacated by the electrons. 

The attractiveness of materials for thermoelectric converters 
can be specified by a figure of merit Z defined as 



Z = 4 



(20.2) 



where T = temperature, °K 

S = Seebeck coefficient, volt/K° (i.e., emf developed per 

unit temperature difference in the specimen) 
r/ = electrical conductivity, (ohm-cm) -1 (i.e., reciprocal 

of resistivity p) 
k = thermal conductivity, watts/ C° cm 



278 Looking In: Atomic and Nuclear Physics 

Both 8 and a depend on the density of conduction electrons in 
the specimen, as shown in Fig. 20M. It is apparent that for 
intermediate- and low- temperature use, semiconductors will pro- 
vide the highest efficiency in thermoelectric converters. Pairs of 
semiconducting comp< ds which have high conversion effi- 



I mutators 



Semiconductors 



Metals 




Fig. 20.2 A thermocouple circuit of dissimilar metals, A 
and 8. The migration of electrons from regions of higher 
density toward regions of lower density produces o con- 
ventional current in the counterclockwise sense. 

ciencies have been found by making binary ami ternary com- 
pounds of materials in groups I, III, and VIj or the periodic 
table: AgSbSe^, CuTiSt, etc. The numerous combinations possible 
make the task of screening and developing the most favorable 
thermoelectric materials a formidable one. Vet exciting progress 




10" 
Electron density, no./cm* 

Fig. 20.3 Properties that govern the choice of materials for thermoelec- 
tric devices, (Courtesy John C. Kelly, VVesfinghouse Research laboratories.) 



40 

- 30 

c 

HI 

9 
a. 
£. 20 

o 

c 
.2 
'o 

LU 

10 



Practical limits of thermoelectricity 



Central station 



esel or marine 




Present ^^ ^S Automobile 



Auxiliary power 

J ! 



-L. 



-L. 



10 100 1000 10,000 

Power rating, kilowatts 



100,000 



Fig. 20.4 Thermoelectric power devices con be competitive with other power 
sources. {Courtesy John C. Kelty, Weslinghouse Research laboratories.) 



279 




Fig. 20.5 Power- producing thermoelectric elements mode of germonium- 
silicon semiconductors. (RCA laboratories, Princeton, N.J,) 



Heat source 
Nuclear 

Nuclear heat transFer 
Fossil Fuel 
^"Waste" heat 

r 



Junction technology 



Thermoelectric 
materials 




Controls 
DC-motching voltage/ 
current inverters 



Fig. 20.6 Thermoelectric power system alternatives. {Adapted from the Genera/ 
Electric brochure GEZ-3Q79B.} 



280 



Unconventional Energy Sources 281 

has been made. Seebeck's original thermocouples (1821) could 
convert heat into electric power with an efficiency of only 2 per 
cent. Study of the PbS-ZnSb couple by Maria Telkes in 1833 
raised the efficiency to 4 per cent. Further work with semi- 
conductors has given the present efficiency of about 17 per cent. 
Theoretical considerations (using quantum mechanics) suggest 
that it will be possible to attain efficiencies as high as 35 per cent. 
This will make thermoelectric power devices competitive with 
existing power sources (fig. 20.4). Each of the germanium-silicon 
thermoelectric elements shown in Fig. 20.5 is capable of gener- 
ating about -i watts upon exposure to heat at about 1000°C. A 
s<]uare-foot platelike arrangement of snch elements could generate 
up to 10 kilowatts, nearly three times the usual electric power 
demand in a home. Some alternatives to be explored in the 
development of a thermoelectric power system are suggested in 
Fig. 20.0. 

20.5 THERMIONIC CONVERTER 

Thermionic emission was noticed by Edison in 1883. In 1956 
V. C. Wilson designed a converter in which electrons are "boiled 




J^Xv. Cooling 



Insulator 





1 "*- ia — 1 — 

Anode 






f Electrons | 

MM* 








Cathode 







Load 



KS Heat 



Fig. 20,7 A thermionic converter. 



out" of a hot metal and used to produce an electric current 
directly. One obvious difference between the thermionic con- 
verter (Fig. 20.7) and the thermocouple is that in Wilson's 
device the metals arc separated by a vacuum or a gas at low 
pressure. There is electrical Row between the electrodes, but there 



282 Looking In: Atomic and Nuclear Physics 

is less flow of heat in this space than through a metal. Thus t ho 
electrodes can be at different temperatures, and the efficiency is 
increased. 

The conversion process is shown in l'"ig. 20.8, where electron 
energy is plotted against distance from cathode to anode. The 
base line corresponds to the energy of the electrons in the cathode. 
Heating the cathode "lifts" some of these electrons over the work- 
function barrier at. the surface of the cathode, w n into the space 
between electrodes. If the electrons can follow path a to the 
anode with only a small loss of energy, there will be a potential 



rs 

I V 
I \ 

I \ b 
I V 
\ 






~~ «^- - < 



Cathode 
> 140Q°K 



Fermi level 

77777777777777^ 



.1 



a 

gas 



^ Plasma drop 

| T Anode ~ 700° 
J" Fermi level 

' Load 



— r 

Output 
voltage 






Fig. 20.8 A plot of electron energy vs. distance (cothode 
to anode) in a thermionic converter. 



difference between the electrodes, capable of doing work in an 
external circuit. In vacuum devices, the electrons entering the 
interelectrode space soon form a space-charge barrier, represented 
by path h. This would increase the cathode electron energy neces- 
sary to electrons to cross to the anode, so the space charge is 
neutralized by adding an ionizable gas, such as cesium. Or 
alternatively a vacuum-type converter is made with a very small 
{0.001 in.) spacing between cathode and anode to minimize space- 
charge effects. 

Current models of thermionic converters are stated by Gear 
eral Electric to have these characteristics: vacuum type, efficiency 
5 per cent, cathode temperature 1100°C; gas-filled type, effi- 
ciency 17 per cent, cathode temperature 15:!0 °C. The gas-lillcd 



Unconventional Energy Sources 283 

unit has the additional advantage of smaller weight per unit of 
power: -! versus 2."> li> kilowatt. 

20.6 MAGNETOHYDRODYNAMICS 

An Mill.) generator utilizes the principle discovered by l-'araday 
that an ion moving in a magnetic held experiences a side push 
(Sees. 10.8 and 10.9), Hot ionized gas is forced between the poles 
of an electromagnet (Fig. 20.9), producing a voltage difference 



To regenerator 



Hot gos Flow 




Flow 
©- 



Field 



Current 



JV 



-VW 



Fig, 20.9 A magnetohydrodynamic generator, 

between the electrodes, at right angles to the magnet. By con- 
necting the elect rudes, power may be delivered to an external 
load. A regenerator is used to recover energy from the emerging 
Kas stream which may still be as hot as 2000°C when its ionization 
has dropped to levels insufficient for effective energy conversion. 
An MUD generator might be operated as part of a conven- 
tional gas or combined gas and steam turbine cycle. Few data 
exist today on which to calculate efficiencies attainable with such 
a combination . Some estimates suggest that addit ion of an M H D 



284 



Looking In: Atomic and Nuclear Physics 



generator could raise tlio over-all efficiency of a generating station 
to 55 per cent. 



20.7 FUEL CELLS 

A fuel cell is a continuous-feed electrochemical device in which 
the chemical energy of reaction of a fuel and air (oxygen) is con- 
verted directly and usefully into electrical energy. A fuel cell 
differs from a battery in that (1) its electrolyte remains un- 
changed and (2) it can operate continuously as long as an externa! 
supply of fuel and air is available. 

Sir William CSrove, an Englishman distinguished in electro- 
chemistry and the law, used a hydrogen fuel cell in his experi- 



Chemieol 
energy 



r 



* Heat 




\ 



*■ Thermoelectric 
-*■ Thermionic * 
*■ Thermogolvanic 



Fuel cell 



/ 



Fig. 20.10 Fuel cells convert chemical energy directly into electric energy, 
thereby avoiding the thermodynamic limitation on the efficiency of heot engines. 

ments as early as 1839. By the end of the last century, Wilhelm 
Ostwald and others came to appreciate, through thermodynamic 
analysis, that the fuel cell is potentially the most efficient simple 
way of converting chemical energy into electrical energy. 

Heat engines are subject to the Carnot limitation of thermo- 
dynamics which says that the maximum theoretical efficiency 
with which heat can be converted into another form of energy is 
determined by the inlet and exhaust temperatures of the engine: 

7> 71 

J i nlet / outlet , on rt\ 



Maximum efficiency = 



7',,, 



i,-i 



It is an attractive feature of the fuel cell that its efficiency is not 
subject to the Carnot limitation, for the energy being converted 
never deteriorates into the random motion of heat. The fuel cell, 
when compared with familiar methods of generating electric 



1 








4) w I 



a.'-* \ f -« 0. 1 



+ 

o 






x 
2 £ 

-2 
I « 
.2 -£ 

K V 



■5 2 



t 



S3 







$MWiW AW Mi 

'" '' IV* 1 'if V 1 '.I: 'r 1 lit. M j ',] 

: 1 \\Ai\i 

I. 1 



^F± 






'v — ^ ,1 I I, I 1 



p 

1 1 "< III 

('"ill 



i! f 'if jf 1 li 1! . MMii , hi 1. if-. 

llil'.i,li.l.l,r l |l.,.il)|l:M l l,'„lt,ll.r l t'^L 



91 

c 
«l 

CD 

>- 

K 

o 



p 



o. 



o 



285 



286 Looking In : Atomic and Nuclear Physics 







Fig. 20.12 A 75-wott 4-cell Allis-Chalmers fuel cell system designed 
for and tested under "jsero gravity" conditions. 



energy (Fig. 20.10), is very direct in its conversion of chem- 
ical energy into electrical energy. Partly because of this incentive, 
fuel cells are probably the most highly developed of the uncon- 
ventional energy-conversion methods discussed in this chapter. 
Under favorable conditions, efficiencies of 80 and even 5)0 per cent 
have been reported with hydrogen fuel. 

A fuel cell, like any other electrochemical cell, contains two 
electrodes: anode and cathode. These are joined externally by a 
metallic circuit, through which the valence electrons from the 
fuel flow, and internally by an electrolyte, through which ions 
flow to complete the circuit (Fig. 20.1 1). These are the electrode 
reactions : 

Anode 2H 2 - 411+ = ee~ 

Cathode Oj 4- 4H++ ier = 2H 4 

Over-all O* + 211, = 2H a O 



Unconventional Energy Sources 287 

The electron does useful work for its in passing from anode to 
cathode in the external circuit. The hydrogen ion completes the 
circuit by going from anode to cathode through the electrolyte. 
The electrons are urged through the external circuit by the 
thermodynamic driving force called the Ciibbs free energy of the 
over-all reaction. 

The major difficulty noted by Grove in 18W) is still a problem 
in the design of fuel cells: how to obtain sufficient fuel-electrode 
(catalyst)-eleetrolyte reaction sites in a given volume. In many 



Fig. 20.13 Unlike other conver- 
sion systems, fuel cells ore more 
efficient at tow output. 



Fuel cell 




50 100 

Rated bod, per cent 



cells, fuel (gas), electrolyte (liquid), and electrode (solid) are 
brought into effective contact by a porous electrode structure 
which depends on surface tension forces to get reasonable contact 
stability. 

In theory a fuel cell can be built in almost any size and 
capacity. Practically, fuel cells are packaged in small modules 
or "batteries" to be connected in series or parallel as needed for it 
particular application (Fig. 20.12). While conventional gener- 
ating devices hei ie less efficient as they so front design load to 

idling, the fuel cell is more efficient at lighter loads (Fig. 20.13). 



APPENDIX 
Reaction Thrust 



The concept of reaction Hi nisi may lie clarified by considering the 
recoil produced by a parallel si ream of particles. Prom Newton's laws 

it follows that for any system <>f objects or particles tin- center of mass 
of the sy-iem moves according to the equation 



F = , mv 
at 



(1) 



where F = net external force applied to system 
m = total mass of system 
v = velocity vector of the center of mass 
/ = time 
Xo matter how complicated the system or how inncli force one of its 
parts exerts on any other, if the net external force is zero (as in field- 
free space), then 



It™ 



il 



(2) 



which stales thai (he total moment urn of Hie system is a constant vector 
i|ii:inl ity. 

Consider a system of two particles, a "rocket" of mass m ami velocity 
and a particle of (jas of mass 8m which is just leaving the nozzle with rcla- 
live velocity t.. The uel momentum of this system is mv + 8m{r — v,). 
From Eq. (2) 



j- [mv + 6m (v — iv)] = 



289 



290 Appendix A 

Rut 8m(di</dt) is negligible, and d(6m)/dt = -ilm ill. since t he mass of 
exhaust gas equals the decrease of the mikei max. Also for the exhaust 
velocity r, . ilr, ill = 0, and m is a small quantity which approaches sera 
in the limit. We have the result 



ilr dm 

m — = — v. 

dl dt 



(3) 



or 



lit 



(4) 



where F is the reaction force on the rocket. The mass flow leaving the 
rocket dm/<lt is re presented by a positive Dumber. The negative sign in 
the equation expresses the fact that Fand '•, are in opposite directions. 



APPENDIX 



B 



Burnout Velocity and Range 



In differentia] notation, Kq. (:i.bs) of A><-. :i7 may be written 

du rfflll m) 



(i) 



Kven if the thrust is not constant-, this equation can be integrated to 
give the velocity !>, at burnout 



«i — i'o = ffu/, In h gtt, cos 8 

flit 



(2) 



If we assume that the rocket starts from rest. c„ = 0, set /. = i.\. n "g, 
and R = Wo/»i&, the ratio of initial mass t" final or burnout mass. 
Kq, (2} may be put in the form 



i'» = '"-tr In R — gtt. eos 9 



(3) 



Here 6, is the duration of burning in seconds. The two averages f, (( and ;/ 
are necessary since the values of both effective exhaust velocity and 
gravitational acceleration are dependent on altitude. 

The altitude reached at burnout for a rocket in drag-free vertical 
flight with practically constant thrust (dttt/dt = const) turns out to be 



h = g„I,k f 1 - p _ . 1 ~ il?«'i + <Vb + Ita 



W 



291 



292 Appendix B 

where Ao is the initial altitude at the start of burning. After burnout, the 
rocket will coast upward to its maximum hoi^Iii h m , Again assuming 
vertical nielli and negligible (has. hut taking into account the variation 
of g with altitude during coasting, (lie masting distance is 



A c = 



>;■ 



(r. + ft*)' 



2{?d r, 5 - t-»*(r, + h b )/2g tt 



(«) 



where r r is the radius of the earth. For a rocket which reaches a summit 
of no more than a few hundred miles, h, is much smaller than r, ami 
\±i\. (5) rednees to a familiar form 



lu M 






(6) 



The summit altitude A„ reached in this vertical flight is 

h m = A* + A c ^j 

To approximate the range <>f a ballistic rocket, one may treat the 
powered portion of the trajectory as vertical ami the coasting portion 
a< elliptic. The coasting range a, along the surface of a nonrotating earth 
has been determined as 



- 2r f sin- 1 



rf 



2g„r, 



-** 



(8) 



The range calculation can be corrected for the earth's rotation by using 
for i>i the vector sum of relative burnout velocity and the velocity of 
the launching site and by adding veclorially to h l the distance the land- 
ing point move's while the rocket is in flight. If %» small, Bq. (8) reduces 
to i he familiar equation for the range of an ideal parabolic trajectory- on 
a Hat earth: 



0u 



(9) 



If r b is large, but less than (2ff r ( )i, t he denominator of Fq. (S) approaches 
zero and s r becomes r f . Hence a burnout velocitv of (2ffur,.)J is just 
sufficient for the rocket to enter into a circular satellite orbit. 

The optimum angle of elevation * »f the trajectory at burnout 
vanes with the desired range * r according to the relation 



tan * - 



1 - sin (s r /2r r ) 



ooa (sJ2r,) 
For short ranges, + = 46", For longer ranges, * is Jess than t.V 



(10) 









APPENDIX 



Schrodinger Wave Equation 



If * is the amplitude of the de ftroglic wave, 

W W ** £ m 

The X in this equation is to be found from the momentum of the particles 

we are discussing. The momentum /' can be related to the kinetic energy 



I mV 



Ei _ ' ^ . JL lir p = V^B, 



in 



2m 



<2) 



The total energy B of a particle is its kinetic energy K, : plus its potential 
energy E„, so 



h a 

~ p~ V2m"c¥^re7) 

and the wave equation becomes 

aso, x-iy ^ity JU-1 
dx- dy* ^^ 1 A* 



(3) 



(4) 



E. Schrodinger showed, in I92(i, that Bohr's rules of quantisation could 
be explained on the basis ( ,f the solutions of this equation. The quantity 
ty (jisi) is called the "wave function" or the "probability amplitude." 
Although + may be negative (or even complex), it turns out that its 



293 



294 Appendix C 

We may ask what sort of eolations this equation would have for 
foe electrons moving m the +x direction. Sinee no forces are applied to 
theeteetrons, they move with constant velocity. Their potential energy 
W the same at all pointe; we may take /' equal to zero. The solution of 
he wave equation in tins ease will be a plane wave, expressive in terms 
of Bmea and cosmos, just as for an electromagnetic wave. 



APPENDIX 



1. BOOKS FOR A PHYSICS TEACHER'S REFERENCE SHELF 

American Association for the Advancement of Science: "The Traveling 
Ilijdi School Science Library." AAAS and National Science Founda- 
tion, Washington, D.C., 1961. 

American Institute of Physics: "Physics in Your High School," McGraw- 
Hill Book Company, Inc.. New York, I960. 

Hrown, Thomas II. (edL): "The Taylor Manual of Advanced Under- 
graduate Experiments in Physics," Addison- Wesley Publishing 
Company, Inc., Heading, Mass., 1959. 

Deason, 11. J. (ed.): "A Guide to Science Beading," The New American 
Library of World Literature, Inc.. New York, 1963. 

Glasstone, Samuel: "Sourcebook on Atomic Energy," D. Van Nostrand 
Company, Inc. Princeton, N..L, 195K. 

Hodgman. CI), (ed.): "Handbook of Chemistry and Physics," Chemi- 
cal Uubber Publishing Company, Inc., Cleveland, 1963. 

[(niton, Gerald, and 1). 11. I >. Poller: ■■Foundations of Modern Physical 
Science," Addison- Wesley Publishing Company, Inc., Heading, 
Mass.. ]!)oS. 

Miehols. W, C. (ed.): 'The International Dictionary of Physics ami 
Electronics," I). Van Nostrand Company, Inc., Princeton, N..L, 
1956. 

National Science Teachers Association: "New Developments in High 
School Science Teaching." Washington, D.C.. I960. I neludes 9-page 
list, "Additional science program materials available." 

Orear, Jay: "Fundamental Physics," John Wiley & Sons, Inc.,, Now 
York^ 1961. 

Parke, N. G.: "Guide to the Literature of Mathematics and Physics," 
Dover Publications, lue., New York, 1958. 

295 



296 Appendix D 

Physical Science Study Committee: "Fhyaes," D. C. Heath and 

Company, Boston, !%(). 
: "Laboratory Guide for Physics," J). C. Heath and Company 

Boston, IWiO. 
Price. Derek John deSollu: ■'.Science since Babylon," Yale University 

Press, New Haven. Conn,, 1961. 
Kesuick, R., and I). Halliday: "Physics for Students of Science and 

Engineering," John Wiley & Sons. Inc., New York. I960. 
Rogers, Eric M,: "Physics for the Inquiring Mind: The Methods, 

Nature and Philosophy of Physical Science." Princeton University 

Press. Princeton, N.J., HltiO. 
Rouse. L J., and R. J. ISarllc: "Experiments for Modern Schools," 

John Murray (Publishers), Ltd.. London, 195(5. 
Weber, R. I.., M. W. White. :iud K. V. Manning: "College Phvsies.'* 

MeCraw-Hil] Hook Company. Inc. New York, 1959, 
White. M. W., K. V. Manning, and R. L. Weber, "Practical Physics," 

McC raw-Hill Book Company, Inc., New York, 1955. Includes 33 

experiments. 



2. SOME PERIODICALS FOR A SCHOOL SCIENCE LIBRARY 

American Journal of Physics. American Institute of Plivsies. 335 Last 

45 St., New York 17. N.Y. 
Xature, Macmillan & Co. Ltd., St. Martin's St., London, WC 2, 

England, and St .Martin's Press, inc., 103 Park Ave., New York 17, 

The Physic* Teacher. American Institute of Phvsies, 335 East 45 St. 
New York 17, X.Y. 

Physics Today, American Institute of Physics, 335 East 45 St., New 
York 17, X.Y. 

The School Science Renew, The Science Master's Association, 52 Bate- 
man St., Cambridge, England. 

Science, American Association for the Advancement of Science, 1515 

Massachusetts Ave., XW, Washington 5, IXC. 

The Science Teacher, Journal of the National Science Teachers Associ- 
ation, 1201 16 St., XW, Washington (i, D.C. 

Scientific American, (15 Madison Ave.. New York 17. N.Y. 
Sky and Telescone, Sky Publishing Co.. Harvard College Observatory, 
Cam I) ridge 3M, Mass. 



3. SOME PROFESSIONAL ORGANIZATIONS OF INTEREST TO THE 
PHYSICS TEACHER 



American Association of Physics Teachers. American Institute. 

Physics. 335 East 45 St., New York 17. N.I . 
Am e rican Chemical Society. 1155 16 St., XW, Washington 25, D.C. 
American Meteorological Society, 3 Joy St.. Boston S, Mass. 



of 



Appendix D 297 

American Backet Society. 500 Fifth Ave,. Xew York 36, N.Y, (Ask for 
latest Book List.) 

American Society for Engineering Education, W. L. Collins, National 
Secretary, University of Illinois, Frbana. 111. 

Astronomical League, 310 Livingston Terr.. BE, Washington 20, D.C. 

Commission on Mathematics, College Entrance Examination Board, 
425 West 1 17 St., New York 27, N.Y. 

Committee on School Mathematics. University of Illinois. Urbana, 111, 

Educational Testing Service, Princeton, N.J. (The Cooperative Test 
Division publishes a loose leaf binder, 805 pp., of Questions and 
Problems in Science, Text Item Folio no. 1. 195ft.) 

National Association of Biology Teachers, Paul Webster. Secretary- 
Treasurer, Bryan City Schools. Bryan, Ohio. 

National Education Association, 1201 Hi St., NW. Washington (i, D.C. 

National Science Teachers Association, 1201 16 St., NW, Washington ft, 
D.C. 

School Mathematics Study Group, Drawer 2502A, Yale Station, New 
Haven. Conn. 

Science Master's Association, John Murray (Publishers), Ltd., 50 
Albemarle St., London, Wl, England. 

Smithsonian Inst it uf ion, Washington 25, D.C. 



4. SOME SUPPLIERS OF PHYSICS APPARATUS FOR TEACHING 

Central Scientific Division, Cenco Instruments Corp., 1700 Irving Park 
Road, Chicago 13. III., and (it Hi Telegraph Rd.. Los Angeles 22, 
Calif. 

The Ealing Corp., 33 University Rd., Cambridge 3S, Mass. 

Macalaster Bicknell Co., 243 Broadway at Windsor St., Cambridge, 
Mass. (Suppliers of PSSC apparatus.) 

Science Materials Center. 5!* [-'mirth Ave., Xew York 3, N.Y. 

The W. M. Welch Scientific Co.. 1515 Sedwick St.. Chicago 10, III. 



5. GREEK ALPHABET 



A 


a 


alpha 


N 


V 


nu 


Ii 


fi 


beta 





£ 


xi 


r 


7 


gamma 








omieron 


a 


a 


delta 


ii 


TC 


l'i 


E 


e 


cpsilott 


P 


P 


rho 


Z 


r 


seta 


V 


a 


sigma 


H 


n 


eta 


T 


r 


tau 


n 





1 liela 


T 


V 


upsilon 


1 


I 


iota 


* 


<P 


phi 


K 


K 


kappa 


X 


X 


chi 


A 


X 


lambda 


* 


* 


psi 


M 


n 


inn 


Q 


w 


omega 



298 Appendix D 



6. SYMBOLS 

= means equal to 

■ means is defined as, or is identical to 

^ means is not equal to 

= means varies as, or is proportional to 

2 means the sum of 

* means average value of j: 
« means is approximately equal to 
> means is greater than "(» means much greater than) 
< means less than (« means much less than) 

" Vof-lln™ {hiir , ° V, ' P <!ifii,) '*&**** fiwt doubtful digit; e.g. 
12,600 ik stated to only three signifieant figures: 1.20 X W 

7. BRIEF INSTRUCTIONS ON USE OF A SLIDE RULE* 

irtwL£?a&' ^P^H? 11 » «««*»% performed on the "C" 
? I »i « S T caes -,? he " um! " ,r J " on the left end of the seale is ,, t |, ( | 

Su£%T££S! ^"l-attherigluemlofthescalei! 

flufe /or multiplication: Set index of "C" seale over cither „f th« 

second fac or on the "C" scale. Read the answer on the •<!>" J.. 
under the ha,rh„e. Determine the location of the decimal point I, "i 
rough mental approximation. ' ■ 

kxamplk: Multiply 17 X 23. See Kg. D.I. 



(lit) Set Left "C" Index 
Over V on "D" Scole 





fo-r 



1 1 ■ ■> -Tr 




Fig. D.l 



Life?.t,v3 



— 



-a: 



i: 



:^r 



^i t ..'.,i i ^i.: > ..;'' | ' , i| ii i ^ i >: t i Vt* ii i i Hbt 



-_. 



(2nd) Under 23 on "C" Scole 
Read 391 on "D" Scale 



//ow (o Dtafe. Division j s generally performed on the "C" and "D" 
scales also. " 

the^^^utS'', 8 -'' ^T 'T 1 ""' ° W tte » um ^tor (dividend) on 
tin- » scale and bring the denominator (divisor) on the "C" scale 

gen*Co. StnU ' tbnS "'"' ' ,,|)VriK!l1 iHu8 *«*ti«M <-"">!esy of Eugene Diets- 



Appendix D 299 

under the hairline. Head the answer on the "D" seale, under the index of 
the "C" scale. Determine the decimal point by rough mental approxi- 
mation. 

EXAMPLE: Divide by 3. See Fig. D.2. 



<hr) Set 3 or." C" 
Over 6 on "D" 



r 



2 



c i : rt ' *? T -V h'K 1 



i- i ■ u 
1 1 1 iJi i 



T 






li I' * 



lT 



(2nd) Under Left "C" Index 
Read 2 on "D" 



Fig. D.2 



flow to Find a 8quctre (too'.. Problems involving square roots are, 
worked on the "A" and "B" scales in conjmietion with the "C" and 
"I)'' scales. Note llial the "A" and "li" scales are divided into I wo 
identical parts, which will he referred to as "A-left" and "A-fight." 

Rule for square roots: If the number is greater than unity, and has an 
odd number of figures before the decimal point, set the hairline over the 
number on "A-left" and read the square root under the hairline on the 
"D" scale. If the number has an even number of figures before the 
decimal point, use "A-right" instead of "A-left." Locate the decimal 
point in the answer by mental approximation. 

If the number is less than unity, move the decimal point an even num- 
ber of places to the ri^ht until a number between I ami 100 is obtained. 
Find the square root of the number thus obtained as explained above. 
To locate the decimal point, move it to the left one-half as many places 
as it was originally moved to the right. 

kxamclk: Find the square root of 507. See Fig. D.3. 



(lsl) Set Hairline to 567 
on Left Ho If of "A" 



^ 



.c i.„.V4.r-.Vi" ".?■%■'■& 

o i 7i ■; ~i h I tTwWWl 



TL 



ii|iiii jm i|i i ii | iii|iii t} i»|i nji| i ; i liK 
> } i : 1 1 1 it i t f 1 1 1 : [ w ; J i w jhii | hii|h 4 hii 1 i ^ ■ » 



i),. - i l " 



\s~rs 



(2nd) Under Hairline on 
"D" Read 238 



Fig. D.3 



300 Appendix D 

l*sc "A4aft," since than is mi odd number of figures before the 
decimal point. By mental approximation, locate the decimal poinl 
after the second significant figure, making the answer 23.8. 

kxamplk: Find the square root of 0.0956. See Fig, ]>.-]. 

(1st) Set Hairline Over 

9.56 on Left Half of "A" 



* '■■•■'•■-.' ...r W..I., .t,„. > . t . > 




D I .1 ■ .1 ■ .1- n n'~a •> n'TT) ~ 



~m 



• • < ■ "■!" ),...i...*..i...j . « p j , t.i.ip 



,k • O im].|.l,,.Jl^,l,l|l. ( .. , 1( , ,ii m i; ■ 



(2nd) Under Hairline on 
"D" Read 3.09 



Fig. D.4 



.Move the decimal point two places to the right, thus obtaining 9.56. 
Fse the "A-left," hecause there is now an odd number of figures before 
the decimal point. Take the square root of 9,r>6. !hen move the decimal 
poiul one place to the left, making the answer 0.309, 



7A. SLIDE RULE BIBLIOGRAPHY 

Bishop, C. ('.: "Slide Rule- How to Fse It." Barnes & Xohle, Inc.. 

New York. 
Bshbaeh, 0. \V., and H. L Thompson: "Vector Type Log Log Slide 

Rule." .Manual no. 1725, llugene Dietisgen Co., 1009 Vine S( 

Philadelphia 7. Pa. 
Naming. M, L:"A Teaching Guide for Slide Rule Instruction," Pieketl 

and Eckel Inc., 1109 South Fremonl Ave,, AJhamhis, Calif 
Harold, Don: "Slide Rule? May I Help . . . ," KeulTel and Baser Co., 

Adams and Third Sis., Hohoken, X.J. 

"Inirnducing the Slide Rule," Wabash Instrument & Specialties 
Company, [tic. Wabash, lud., 1943. 

"Ii'- Easy to Use Four Post Slide Rule." Educational Director, Freder- 
ick Post Co., 3050 North Avondale Ave., Chicago, 111. (A projec- 
tion slide rule is expected to be available soon for classroom use.) 

Johnson. L. H.: "The Slide Rule," I). Van Xostrand Company. Inc., 
Princeton. X..1. 

-Macliovina. P. K.: "A Manual for the Slide Rule," McGraw-Hill Book 
Company, foe, 330 Weal 12 St.. New Ym-k 30, N.Y., 1950. 

"Mathematics, .Mechanics, and Physio." Engineers Council for Pro- 
fessional Development, 29 West 39 St., New York, N.Y. 






8. TRIGONOMETRIC FUNCTIONS 



Radians 


[;•■:.]'.•.■'. 


Sine* 


Cosine? 


Tang ants 


Cotangents 






.0000 


a 


.0000 


1.0000 


,0000 


s 


00 


1 . 5708 


.0178 


i 


.0175 


,9988 


.0175 


57.20 


80 


1 . 5533 


.(Will 


2 


.0349 


.9984 


.0340 


28 . 04 


88 


1 ,5359 


0524 


if 


.0523 


.9986 


.0521 


10,08 


87 


1.5184 


,0698 


4 


.0008 


.9976 


.0699 


14.30 


86 


1 5010 


.0878 


5 


.0872 


.9962 


.0875 


11.430 


85 


1 . 1835 


Ml 17 





. 1046 


.9946 


.1051 


n 61 i 


84 


1 . (661 


. 1222 


7 


1219 


. 05)25 


. 1228 


8.144 


83 


1.4480 


. 1390 


8 


.1392 


mod3 


.1 105 


7.115 


S2 


1 .4312 


.l, r >7] 


9 


. 1504 


.9877 


. 1581 


0.314 


81 


1.4137 


.1745 


10 


.1736 


.9848 


. 1703 


5.671 


SO 


I 3963 


.1920 


11 


ions 


.9810 


.1914 


5 ! 15 


79 


1.3788 


.2091 


12 


,21)70 


.978! 


.2120 


4.705 


78 


[ .3614 


2209 


13 


.2250 


.9741 


.2300 


4.332 


77 


1.3439 


21 !3 


14 


.2410 


.9703 


.2493 


I Oil 


70 


1.3205 


.2(il8 


15 


2588 


0059 


.2070 


3.732 


75 


1.3090 


.2798 


16 


•J 7511 


.9613 


.2867 


3.487 


74 


I .2918 


. 2967 


17 


.2021 


, 9563 


.3057 


3.271 


73 


1.2741 


.3142 


18 


.3090 


.9511 


.3240 


3.078 


72 


1 .2666 


,3310 


19 


.3250 


.0455 


,3443 


2 904 


71 


1.2302 


.3491 


20 


.3420 


.9397 


.3640 


2.748 


70 


1,2217 


, 3605 


21 


3584 


.9336 


.3839 


2.606 


69 


1.2043 


. 38 H> 


22 


.3746 


9272 


.4040 


2 175 


08 


1.1808 


. 101 1 


23 


.3907 


'.1205 


. 4245 


2.350 


67 


1.1694 


.-1189 


21 


.4007 


.9136 


.4152 


2.246 


«a 


1.1519 


. 4363 


25 


.4220 


.9003 


.4003 


2 144 


05 


1.1345 


.1538 


211 


.4384 


.8088 


,4877 


2.050 


< 1 


1.1170 


.1712 


27 


. 4540 


.8010 


. 5095 


1 . 963 


68 


1 .0990 


.4887 


2S 


1695 


.8829 


.5317 


1.881 


02 


1.0821 


.5081 


29 


IMS 


.8740 


.5543 


1.804 


01 


1. O047 


5236 


30 


.5000 


. 8660 


.5771 


1.732 


00 


1 .1)172 


.5411 


31 


5150 


S572 


.0009 


1.664 


59 


1.0297 


.5586 


32 


.5299 


S1S0 


0249 


1 .000 


58 


1.0123 


.5700 


33 


.5440 


.8387 


.0494 


1 54(1 


57 


0.9948 


. 5934 


34 


5592 


.8200 


. 0745 


1 is:; 


56 


0.9774 


Jill)!) 


35 


.5736 


.8102 


7002 


1.428 


55 


0.9599 


.6283 


: j ,r, 


.5878 


SOOt) 


7266 


1 370 


54 


(1 1)125 


.0458 


37 


.0018 


.7980 


. 7530 


1.327 


53 


0.9250 


.6632 


38 


11157 


7SS0 


.7813 


1 . 280 


52 


0.9076 


.0807 


39 


.112!):! 


.7771 


.8098 


1.235 


51 


o 8901 


.0081 


40 


lltiS 


. 7660 


. 839 1 


1 . 192 


50 


0.8727 


.715(1 


II 


.0501 


.7547 


8693 


1 . 150 


49 


0.8552 


. 7330 


42 


.0091 


.7431 


.9004 


I. Ill 


48 


0.8378 


. 7605 


43 


.0820 


. 73 1 1 


,9325 


1.072 


47 


0,8203 


. 7079 


44 


.0047 


. 7 1 93 


.0057 


1.030 


46 


0.8029 


. 78, r )4 


45 


.7071 


.7071 


1 0000 


1.000 


45 


0.7854 






Coiinei 


Sinfti 


Colon genii 


Tongont? 


Degree* 


Rudiufis. 



301 



302 Appendix D 



9. LOGARITHMS TO THE BASE s 

These two pages give the rial lira! (hy- 
iierbohe, or Napierian) logarithms of tnnii- 
bera between I and m, correcl in four 
places. Moving the decimal point » places 
to the right (or left) fa tfie number is 
equivalent to adding ,, limes 2,:<02(i (or n 
tunes 3.6974) to the logarithm. 



1 

'2 
3 
•1 
5 
6 

i 

8 




2.8026 

i 6052 

6 B078 

8.2103 

11.6129 

13.8165 

IS 1181 

18.4207 

20.7233 



I 
2 

3 
4 
G 

6 

7 
8 
9 



1.0 

i.i 
i.j 
i. a 

1.4 

l.S 
1.6 
1.7 
l.S 
1.9 

2.0 
2.1 
J.J 
2.3 
J.4 

J.S 

2.0 
J.7 
2.8 
2.6 

3.0 
3.1 
3.2 
J.S 
3.4 

3.5 
3.0 
J.7 
3.8 
S.9 

M 

4.1 
4.2 
4.3 
4.4 

4.5 
4.0 
4.7 
4,6 

4 LI 



0.00OQ 
0953 
1823 
1624 
3305 

4055 
470(1 
5300 

5S7.V 

0419 

0.8931 
7419 
7885 
8329 
8755 

9ISJ 

9555 

0.9933 

1.0296 

0647 



0100 0108 iiair, 0392 

1044 1133 1222 1310 

IMS J070 2151 

2770 2852 2927 



mm 
2700 



1314 
1632 
10811 
•>•>$<■ 

2528 
1809 
3083 
3350 
JO 10 

1.3883 
4110 
436! 

!>,, 

4810 

5041 
6201 
5476 
5688 
6892 



1436 3607 3577 3640 

4121 4187 4253 4318 

4762 4824 4836 4947 

53S5 5423 6481 SS39 

5933 5988 8043 6098 

8471 0523 6675 6627 

698t 7031 7080 71JS 

7467 7514 7561 7608 

7930 797S 8020 8065 

S372 8416 8450 850J 

8700 8838 8S70 8020 

9203 0243 92*2 9322 

9504 9632 0670 9708 

9960/0006 0043 0080 

033J 0367 0403 0438 

0682 0710 0750 0784 

1019 1053 1086 1119 

1346 1378 1410 I44J 

1063 1094 1725 1756 

1009 2000 2030 2060 

2267 2206 2320 2355 

2556 2585 2613 2641 

2137 2S«5 2M)2 2M| 

3110 3137 3164 3191 

3376 3403 3429 3455 

3635 3661 3680 3712 

3888 3013 3938 3962 

4134 4159 4183 4J07 

4375 4398 4422 4446 

4600 4633 1666 4670 

4839 4861 4884 4907 

5063 5085 5107 5120 

5282 5304 5320 6347 

5497 5518 5530 5560 

5707 5728 5748 5769 

6013 5033 5953 6974 



0488 0583 0077 0770 0862 

1398 1484 1570 1055 1740 

2231 2311 2390 2469 JS46 

3001 3075 3148 3221 3293 

3710 3784 3853 3920 3088 



43S3 
5008 
.i,W, 
6152 
6678 

7178 
7665 
8109 
8644 
8901 

9361 
0746 

0111, 

0473 

IIS IS 

1161 
1474 
1787 
2000 
J3S4 

2669 
2947 
3216 
3481 
3737 

3987 
4281 
4469 
4702 
4929 

51SI 
6309 
5581 
5790 
5994 



4447 4S11 4574 4837 

5068 5IJ8 SI8S 5J47 

56.53 5710 5766 5822 

6208 0259 0313 0366 

0729 0780 6831 6SS1 

7227 7275 7324 7372 

7701 7747 7793 7839 

8154 8198 8242 8286 

8587 8829 8671 8713 

0002 9042 9083 9123 

9400 9439 9478 gsi7 

9783 9821 98S8 9806 

0152 0188 0225 0260 

0508 0543 0578 0513 

0952 0886 0918 0953 

1184 1217 1249 1282 

1506 1537 L500 1600 

1817 1B48 1878 1009 

21 19 2149 2170 JJ08 

24 13 J442 2470 2499 



MM 

2976 
3244 
3507 
3702 

4012 
4255 
4403 
47J5 

4!>. r ,l 

5173 
6390 
5602 
S8I0 
6014 



2720 2754 2792 

2002 3020 3056 

3271 3297 3324 

3633 3558 3584 

3788 3813 3838 

4036 4001 4085 

4279 4303 4327 

4510 4540 4563 

474s 4770 471)3 

4074 4090 5019 

5195 5217 5230 

5412 5433 5464 

5023 5844 5865 

5831 5851 5872 

8034 8054 8074 



19 

0.0053 
1823 
2024 
3365 
4055 

4700 

5306 

5878 

6419 

0.6931 

7419 
7885 
8320 
8755 
9103 

9565 
0.9g33 
1.0296 

0647 
1.09S6 

1314 
1632 
1939 
2238 
2526 

2800 
3083 

3350 

3610 

1 3m;;i 

4110 
435) 
4586 
4816 
6041 

5261 
5476 
5680 
6802 
1.6094 



6974-3 

'.HUH ."i 

05)22-7 

7897-10 

4871-12 

I Sir, I I 

8819-17 

6793-19 

27(17-21 



Tflnrhf of the 
Tabular 

Difference 
12 3 4 5 

10 10 29 38 48 

17 20 35 44 

8 15 24 32 40 

7 15 22 30 37 

7 14 Jl 28 34 

13 19 26 32 

8 12 18 24 30 
8 II 17 23 J8 
5 II 18 22 27 
5 10 15 21 26 

5 10 15 20 24 
5 9 14 19 23 
4 9 13 IS 22 
4 9 13 17 21 
4 8 12 16 20 



8 II 10 20 

8 II 15 19 

7 11 15 18 

7 It 14 18 

7 10 14 17 

7 10 13 18 

6 10 13 16 

6 II 15 

6 9 12 15 

6 9 12 14 



8 II 14 
8 II 14 
8 II 13 
8 10 13 
8 10 13 



2 5 7 10 12 

2 5 7 10 12 

2 5 7 II 

2 5 7 9 11 

2 4 7 9 11 

J 4 7 9 II 

2 4 6 9 II 

J 4 8 8 11 

J 4 8 10 

2 4 6 8 10 






Appendix D 303 



LOG, (BASE e = 2.718284) 






















Tenrht 


















of the 


















Tabular 


















Difference 




9 


1 I 


1 4 


5 


6 7 8 


9 


10 


1234 5 


5.0 


1.8094 


0)14 6134 


0154 0174 


6194 


0214 6233 6263 


0273 


6292 


2 4 6 8 10 


G.1 


112! 12 


8311 6332 


6351 6371 


6390 


8409 8429 8448 


6487 


6487 


2 4 6 8 10 


8.2 


6487 


6508 66J5 


6544 6563 


8582 


6601 6620 6630 


6668 


6677 


2 4 6 8 10 


S.J 


6677 


0696 0715 


6734 0752 


6771 


6700 6808 6327 


6845 


6864 


2 4 6 7 


6.4 


6864 


8882 8901 


6910 6938 


MM 


6974 8993 7011 


7029 


7047 


14 6 7 9 


6.5 


7047 


7060 7084 


7102 7120 


7138 


7156 7174 7192 


7210 


7128 


24 57 


6.6 


7228 


7246 7J63 


7281 7199 


7317 


7334 7351 7370 


7387 


7405 


2 4 5 7 9 


6.7 


7406 


7422 7440 


"457 7475 


7492 


7609 7527 7544 


7561 


7579 


2 3 5 7 9 


6.8 


7679 


7690 7813 


7630 7647 


7864 


7681 7099 7716 


7733 


7760 


2 3 5 7 9 


5.9 


7760 


7760 7783 


7800 7817 


7834 


7851 7867 7884 


7901 


1.7618 


2 3 6 7 8 


CD 


1.7016 


7934 7951 


7967 7984 


8001 


8017 8034 8050 


8060 


8083 


2 3 6 7 8 


6.1 


8083 


8090 8116 


6132 8148 


8185 


8181 6197 82)3 


,j,,l 


8245 


2 3 5 7 8 


6.2 


6245 


8262 8178 


8294 8310 


8326 


8342 8358 8374 


8390 


-41')-, 


2 3 5 8 


6.3 


8405 


8421 8437 


8453 8460 


S4S", 


8500 8516 6532 


8647 


8563 


2 3 5 8 S 


6.4 


8563 


8579 8594 


8810 6025 


8041 


silSft K67J v ( ;s: 


8703 


8718 


13 5 6 8 


6.5 


8718 


8733 8749 


8784 8779 


8795 


8810 8825 8840 


MM 


8871 


2 3 5 S 


6.6 


8S71 


8886 8901 


8918 8931 


-8940 


8961 8976 8991 


9008 


0011 


2 3 5 6 8 


0.7 


9021 


9036 0051 


'.Win; iw-ii 


0095 


9110 9125 9140 


9155 


9109 


13 4 7 


6.8 


0109 


9184 9100 


9213 9228 


9142 


9257 9272 9280 


9301 


9315 


13 4 6 7 


6.9 


0315 


0330 9344 


9369 9373 


93S7 


9402 9410 9430 


9445 


1.9459 


13 4 7 


T.O 


1.9459 


0473 948S 


9502 0516 


0530 


9544 9560 9573 


9587 


9601 


13 4 6 7 


7.1 


96UI 


0615 0629 


0643 0857 


0671 


0686 9600 9713 


9727 


0741 


13 4 7 


7.1 


9741 


9755 0769 


emg »th 


0810 


0824 0838 9851 


9385 


1 ',-7'.' 


13 4 8 7 


7.3 


1J870 


9802 9900 


9920 9933 


9947 


998) 9974 0B88/0O0I 


2.0015 


13 4 5 7 


7.4 


2.0015 


0028 004! 


0055 0069 


0082 


0000 0109 0122 


OI30 


0149 


13 4 5 7 


7.5 


0149 


0102 0176 


(1I.1SI B202 


0215 


0229 0241 0255 


0268 


0281 


13 4 5 7 


T.fl 


0281 


0295 0308 


0311 0334 


0347 


0360 0373 0386 


0399 


0412 


13 4 5 7 


7.7 


0412 


0425 043K 


0461 0464 


0477 


0490 0503 0516 


0528 


0541 


13 4 5 1 


7.8 


0541 


0554 0567 


0580 0592 


0805 


0618 0831 0643 


0656 


0660 


13 4 5 6 


7.9 


IlW'.i 


0681 0604 


0707 071!l 


0732 


0744 0757 O70II 


0762 


2.07114 


13 4 6 6 


1.0 


2.0704 


0807 0810 


0832 0844 


0867 


OS00 0882 0894 


0906 


ll'.llll 


12 4 5 6 


8.1 


0010 


0031 0843 


0058 096s 


0080 


0002 1005 1017 


1020 


1041 


12 4 5 6 


S.2 


1041 


10S4 1080 


1078 1090 


1101 


1)14 1126 1138 


1150 


1163 


12 4 8 6 


8.3 


1163 


1175 1187 


1199 1211 


1223 


1235 1247 1158 


1270 


1282 


12 4 5 5 


8.4 


1282 


1204 1300 


1318 1330 


1342 


1353 1305 1377 


13SH 


1401 


12 4 6 6 


8.S 


1401 


1412 1424 


1436 1448 


1469 


1471 1483 1494 


1506 


1618 


12 4 5 6 


8.6 


1518 


1529 1541 


1562 1564 


1576 


1587 1590 1610 


1822 


1623 


12 3 5 6 


8.7 


1833 


1645 1660 


1668 16711 


1601 


1702 1713 1726 


1730 


1748 


12 3 5 6 


J .8 


1748 


1759 1770 


1781 1793 


1804 


1815 1827 I83S 


1849 


1861 


12 3 5 6 


8.9 


1861 


1872 1883 


1894 1906 


1917 


1028 1039 1050 


1961 


2.1972 


12 3 4 8 


J.O 


2.1971 


1083 1904 


2006 1017 


2028 


2039 2050 2061 


2072 


2083 


12 3 4 6 


9.1 


2083 


2094 2105 


2116 2127 


1138 


2148 2159 2170 


2181 


2102 


12 3 4 5 


1.2 


1192 


2103 2214 


2225 2235 


2240 


2257 2288 2270 


2 2 Mi 


2300 


12 3 4 5 


9.3 


2300 


2311 2322 


2332 2343 


2354 


2364 2376 2386 


2396 


1407 


12 3 4 5 


9.4 


2407 


2418 2428 


2439 2450 


2460 


2471 1481 2492 


2502 


2513 


12 3 4 5 


9.5 


2513 


2523 2534 


2544 1.555 


2565 


2576 2586 2597 


2607 


2616 


1 2 3 4 F 


9.8 


2'Us 


2628 2638 


3649 21159 


2670 


2680 209O 2701 


2711 


2721 


12 3 4 6 


S.7 


2721 


2732 2742 


2752 2702 


2773 


2783 2703 2803 


2814 


2v>4 


12 3 4 5 


0.8 


2824 


IS34 2844 


2854 2865 


2875 


2886 2895 2906 


1915 


2 '.■:•.'. 


1 2 3 4 S 


9.9 


2925 


2935 1948 


2966 1966 


2978 


2980 2996 3006 


3016 


2.3026 


12 3 4 5 



304 Appendix D 



10. VALUES OF PHYSICAL CONSTANTS 

As experimental data improve, "best values" of the physical constants 
are recomputed by statistical methods. See, for example, K, It, Cohen, 
J. \V. M. Do Mond. 'I". \V. Lay Ion, and J. S. Hollelt. "Analysis of 
Variance of the 1052. Data on the Atomic Constants and a New Adjust- 
ment, 1885" Review of Modern i'hysv-s, 27:303 380 (1955). The values 
listed below have been rounded off from those liste<l in the paper cited 
and have been expressed in inks units. The physical scale is used for 
all constants involving atomic 0)88868. 

Avogadro's number: A'.i = 6.0249 X 10 10 molecules kmole 
Gas constant per mote: R« = 8,31 7 joules/(kmoie)(°K) 
Standard volume of a perfect gas: V a = 22.420 m 3 atm. kmole 
Standard atmosphere: i>» = 1.013 x 10* newtons/m* 
Speed of light in free space; c = 2.9979 X 10" m/sec 
Electronic charge: e = 1.0021 X 10- |!l coul 
Planck's constant: A = 6.6252 X SO" 3 " joule-sec 
Faraday constant: F = 9.652 X 10 7 coul/ kmole 
Charge/mass ratio for electron: e/m = 1.758" X 10" coul/kg 
Rest mass of electron: m = 9.1083 X 10~ 31 kg 
First. Uohr radius: «„ = 5.2917 X 10 ll m 

Compton wavelength of electron: X = h/mc = 24.203 X 10 13 m 
Boltamann's constant: A.- = 1.3804 x 10 S3 joule, °K 

= 8.617 X10 B cv/"K 
.'•lass-energy conversions: I kg = 5.610 X 10 s * Mev 

1 electron mass 0,51098 Mev 
1 proton mass = 938.21 .Mev 
1 amu = 931.14 Mev 

1 neutron mass = 939.51 Mev 
Energy conversion factor: I ev = 1.6021 x 10 "joule 
Rest masses: electron m = 9.1083 X 10 31 kg = ft.4870 X 10 • amu 

proton »t„ = 1 .0724 X 10"" kg 

neutron »i„ = 1.6747 X 10 - " kg 
Proton mass electron mass ratio = 1,830.12 
(iravitational constant: = 6.67 X 10 ll newton-mVkg* 



Appendix D 305 



11. CONVERSIONS OF ELECTRICAL UNITS 



Quantity 


Symbol 


Practical 
unit, mfcs 


Cgs-esu equiv. 


Cgs-emu equiv. 


Energy 

Current 

Electronic potential 

Electronic field 

Magnetic flux 

Magnetic induct. 

Permittivity of free 


w 

( 

V 
£ 

a 
B 


1 jouJe 
1 ampere 
1 volt 
1 volt/m 
1 weber 
1 weber /m 2 
BBS X 10" 11 

coulV 

newton -ni ! 

1.257 X 10" 6 
newton/ 
omp- 


10" ergs 

3 X 10 9 jtotomp 
J X 10 -s statvolt 
10~ 4 iv/cm 

statcoulomb 
dyne cm 5 

1 1 

9- 10" V*wso 


10" ergs 

0.1 abomp 
10* ob volts 
1 f ' abv/cm 
10^ maxwells 
10' gauss 
1 1 


space 

Permeability of free 
space 


9 ■ 10" V«M> 

unll pole 
dyne cm 5 


Note: / — 


= C 









Index 



Absorption coefficient, 265 
Acceleration cine to gravity, 41, 43 
Acoustic waves, 1(11, 103 

(See a/so Wave) 
Activity, 240 
specific. 242 
Adams, ('. C, 7. 8 
Actlu i- theory, 139, 1st; 
Ampere, 110 

Ampere's law, 131, 150, 161 
Amplitude, !l(i, 106 
Angle of ascending node, 51 
Angle of inclination, 51 
Angular momentum, 55 
Anode, 140 
Apogee, BJ 

Apparatus, suppliers of, 297 
Argument of perigee, 53 
Wending node, angle of, 51 
Asteroids, 10, 13 
Astronautics, bibliography of, 7 
careers in, 7 
lilms on, 7 
history of, 3 
Astronomical unit, 9 
Atmosphere, entry of, 72 
Atom, 7!i 

earlv concept of. 78 
models or, 117. 203 205. 23\ 25:1 
radius of, K5 
speed of, 86 
Atomic Knergy Commission, 32 
Atomic muss, 70, 153 
Atomic mass unit, 70 
Atomic number, 118, 183, 205, 211, 

220 
Atomic weight, 79 
Auger effect, 175 

307 



Autoradiography, 203 
Average life, 24 1 
Avogaclro's number, 78, 144 

Baker, K. 11.. IS, 19 

Bnlmcr Series, 208, 212, 214 

Hand theory of conduct inn. _M-"> 

Harkla, C. G, 281 

Barrier, energy, 220, 244 

Bertie, R. J., 290 

Bauer. 0. A.., IS 

Beats, 197 

BecquereJL H. A., 77, 203 

Benson, O. ()., 7 

Bernoulli's theorem, I'll 

Beta decay, 210 

Bel si spectrum, continuous. 24S 

Binding energy, 299, 242. 251 

Bishop, C. ('.. 390 

Blaekbody, 168 

Blackbody radiation, 166 

energy distribution in. lit) 

(See also Radiation) 
Blunchard. C. II.. 153 
Bohr. N.. 1 18, 293 
Bohr atom model, 293 204,297.222 

(See iitao Atom) 
Hook list. 205 
Bore, 08, 9!) 
Hot he. W.. 253 
Bragg diffraction, electron analogy 

of, 21 S 
Breeder reactor, 271 
Breillat rahluug, 17li 
Brown, T. IS., 295 
Bubble chamber, 203 
Budierer, A. II.. 198 
Bnchheiin, It. YV., 7, 70 






308 Index 

Burnett, C. It., 163 

Burnout .speed, rocket, 20, 201 

{See alxo Rocket ) 

Cajori, !■'., 64 

Calorie, ss 

Careers in astronautics, 7 
[See also Astronautics) 
Curncit efficiency. 32. 2x4 
Cathode, 140 * 
Cathode ray tube, 143 
Cal lioiie rays, 1 30 
Cavity radiation, L89 
Center of mass, 20 
^ rotation about, 212 
Chad wick. J., 253 
Chain reaction, 274 
Charge, of an electron, 145 
Charge /mass ratio, 127. 142 
Charging, 117 
Chromosphere, II 
Circular orbit, 46, 53 
Clock paradox, 195 
Cloud chamber, 263 
Cohen, ]■:. H., :«)4 
Collisions, molecular, 86 

nuclear, 252 
Comets, 11), i:j 
Common ell erf, 176, 265 
Condon, E. U., 210 
Conduction of elect rieity in a tas 

139, 140 
Conductivity, band theory of, 235 
electrical and thermal. 233 
of metals, 146 
quantum theory of, 234 
Conic orbits, 65 
Conic sections, 64 
Conservation of energy, 200, 244 
Constants, physical, :mm 
Corona, 1 1 

Correspuiidence principle, 183, 214 
J osmic rays, II, 14, 208 
Coulomb. I It) 
Coulomb barrier. 220. 244 
Coulomb's law, lis, 200 
Cou titer, 201 

(leiger-MuUer, 261 
scintillation, 202 
Crew, II., mi 

< 'fit teal mass, 27 1 

Cross product, vector, 56 

Curie, 242 

Current, conventional, 127 

direction of, 12s 

electric, (27 

electron, 127 

induced, 136 

in magnet ie field, 130 

in metals, Hti, 235 



Daltnn, J.. 77. 7.s 
Damped wave, 106 

Darwin, C, R. 115 

Davidson, U'. ].., 273 

Davisson ami Germer cxia-riim-nt 

2I.S 
Deason, H. J., 205 
De BrogUe, I,., 216 
De Broglie wave, 170, 222, 230 
Decai constant, 211, 25<J 
Deflection of charged particles, 125, 

Degree of freedom, S8 

I 'elector, radiation, 2til 

Do ^'auvenargues, Marquis, 138 

Dewey, John, 40 

Dill ruction, 107, ISO 

of electrons, 2 IS 

of neutrons. 210 

of photons. 2 IS 
Diffraction grating, 108 
Dilat atio n of lime, I it 
Dirac, P. ,\. M., 216 
Direction rules, for induced emf, 
136 

for magnetic force, 134 

for magnetic induction, 134 

Disintegration energy, 243 

I Kspucement, 06 
Disraeli, Benjamin, lis 

Distances, to planets. HI 

to stars, IS 
Dobie, .1. [''rank. ti7 
Dopplcr effect, 112, 107 

transverse, 1 14 
Dose, radiation, 267 
Dunne and Hunt law, 170 
DnClaux, lit) 

Dulongand IViit law, I4S 
DuMoml.,1. U\ M., :«)4 
Duncan, .1. c. mi 

Dyne. 21 

Earth satellite, fit 

Eccentricity, .",1 

Eddington, A. 8., 115. 271 

Edison, T. A„ 2X1 

Effective exhaust velocity. 25, 30 

Efficiency of heal engine', 32. 284 

Einstein, A., !t, 67, 77, 173, 175, 

1X4, 104, 196 
Einstein's mass-energy equation, 

199 
Einstein's photoelectric equation, 

173 
ESectiio current, 127 
in magnetic Meld, 130 
in metals, 148, 235 
Electric field intensity, [21 
Electric potential, 122 






Electrification, 1 17 
Electrolysis, I It 
Electromagnetic wave, 103, 161 

energy of, (66 

gamma ray, 242 

plane, 103. (66 

speed of. 104 

x-ray, 175 
Electron, 10s. 127, 130, 144 
charge or, 145 
and electrolysis, 111 

e/m ratio for, 142 

energy data for. Hi!) 

free, 146, 232 

in nucleus. 246 

shells, 227, 230 
Electron configuration, 228, 230 
llecfroti (iilTraetion, 2 Is 
Electron (low, 127 
Electron theory of conduction, 146 

Section volt, 124 

Hectrostatic units, 1 19 
demerits, periodic table of, 228 
Elements of an orhil, 51 
Jliot. C, 166 
jllipse. 61, 64 
Emerson, It. W., 77, 1S3 
imf, induced, 135 
Energy, binding. 20(1, 254 

conservation of. 21 Ml, 244 

disintegration, 243 

distribution in blacklmdy radia- 
tion, 170 

in electromagnetic wave, 105 

oquipnrtilion of, SS 

forbidden, 237 

in nuclear reactions, 243, 25S'. 
273. 270 

potential, 44 

quantization of, 2(H), 220 

of satellite. 1 1 

sources of, 272 

tiergv barrier. 220, 244 

uerg'y levels, 2IIX. 21(1, 235. 242 

qua! ion of state. 81 

quilihrium, radioactive, 250 

quipnrthinn of energy, 8S 

scape speed, 31, 45 

shlmch, O. W., 301) 



xelusion principle, 227 

Ixlinust velocity. efTeclive, 25, 30 

Exponential law of decay, 230 



Falling body, 42 

Faraday, ML 136, 144, 150, 283 

Faraday, 144 

Paradays law of induct ion. 158, 

Kit I 
Fermi, K., 273 
Fermi distribution, 234 



Index 309 

field, electric, 121 

deflection of particles bv, 125, 
Ml 

gravitational, 42 

magnetic, 125 

deflection of particles by, 125, 
120, 141 
Field intensity, electric, 121 
Field strength, magnetic, 105 
Film lists, 7, IS. 37 
Fission, nuclear, 273 
Fitzgerald-Lorents contraction, 104 
Flux, electric, 157 

magnetic, 135 
Forbidden energies, 237 
Force, gravitational, 41 

magnetic. 134, 142 
on a current, 120 

nuclear, 206, 254 
Fourier series, 07, 104 
Founder. (I,, 201) 
Franklin, YV. S., 138 
Free electrons, 140. 232 
Free fall. 43. 4li 
Freedom, degree of, 88 
Frequency. 114 
Fuel cell, 284 
Fusion, unclear. 274 

a, gravitational acceleration, 41, 43 
(7, gravitational constant, 41 
Galaxies, 17 
Camilla decay, 242 
Gas, fully ionized. 275 

ideal, 80 

kinet ic I henry of, S! 
Gas constant, so 
Gas discharge tube, 140, 141 
i las law, so 
Gauss, 120 
Gauss's law, 157. 150 
( lav-Iaissac, .1.. 7s 
Geiger. H., 203 
(ieiger-Mnller counler, 261 

rator principle, 186 
Glaser. D. A,, 263 
(ilasstone. S., 205 
Goddard, H. II., 4 
Gram tuoieciilar volume, 70 
Grating, optical, 108 
Gravitation, universal, 41 
Gravitational acceleration, i/, 41 

on planets. HI 

slandard, 25 
Gravitational constant, 0, 41 
( iravitat iunal field. 42 
Gravitational force, 41 

on planets, 10 
Gravitational potential energy, 44, 
47 



310 



Index 



Gravity. 41 

( Irsek alphabet, 207 

Croup velocity, l(J(l 
Croups, orbital, 227, 230 
drove, W., 284 
(iuidnnco of rocket, "(I 

Halm, a. 273 
Half-life, 241 

(See (i/.to Radioactivity) 
Hall effect. 233 
Ilalliday, I).. 2!lli 
Hartung. M, I„. 300 
Heat engine, ellieienev of, 32 
Ilcavisido, (I., 120 
Heiscubcrg, W,. 2 It; 
Hciscnljerg a uncertainty principle, 

182 
Henry, J.. 135 
Herald, ]>., 300 
H err iik. K., 02 
Hertz, (i.. 157. KM 
High-ejicrgv particles. 260 
Hohbs. M„ 3S 
Hodgroan, (". 1),, 2!>f> 
Hull on, (i., 205 
Horace. Ill 
I lo vie, V., I!) 
Hubble, !■:. P.. go 
Huxley. T. H., 252 
Hydraulic jump, SIM 
Hydrogen, Hulir model of, 207 

energy levels for, 208, 210 

spectrum of, 207 

(See alao Atom) 

Ideal ga« law, SO 
I (u puke, specific, 25, 27, 32, 33 
Inclination, angle of, 51 
Induced current, 135 

direction of. 135 
Induced end. 135 

direct ton of, 138 

Induction, magnetic, 128, 131 

at center of loop. 132 

direction rule for, 134 

Faraday's law of, IjjK 

force due to, 134 

near Straight wire, 133 
Insulator, 237 
Intensity. ]{I5 
wave, 165 

(See alto Wave) 
Interference. 106, 1*7 
Interferometer. iMi 
Interplanetary travel, OR 
Ion. 144. Mi) 
Ion propulsion, 34, 3d, 71 
Ionization chamber, 201 
Ionization energy, 229 



Ionization potential, 2in, 211 

Ionizing radial ion, 2(51 

iHHtojws, 140, I />:* 

Ives. II. i:.. Ml 

Ives and Stilwell experiment, li)7 

James, J. N., m 
Jet .separation, 2!l 
Johnson, L. 11., 300 
Joliot. 1''.. 253 
Jupiter, lit, 12 

A*-eapture, 217 

Kelvin, Sir William Thomson, 271 

Kepler's laws, 56, (HI 

derivation of, (i2 
Kinetic theory of gases, 81 
Kirchhoffs law, lli.s 
Kiwi engine, 32, 33 
Krogdahl, W. S.. I!) 

l-i ert, B., St!. S7 

Launching speeds, 70 

(See also space) 
Layton. 1'. W., 304 
Lens's law, 135 
Life, on planets, is, 10 

of radioisotope, 241 
Light, speed of, |(>4, |8(i, 1JI2 
Line of force, gravilat imial, 42 
Loeb, L. I).. s:i 
Logarithms, 3112 
Lorents, H. A., 232 
Lorentz transformation. 193 

Much number, 2s, 102 

Machoviaa. !'. K.. 300 
McLaughlin. !l. ».. Ml 

Magnetic deflection, I3ti, 141 
Magnetic field strength. I(i5 
Magnetic Mux, 135 
Magnetic induction, I2K, 131 

ill center of loop. 132 

direction rule for. 131 

Faraday's law of, 158 

force due to, 134 

near straight wire. 133 
M.'ignclohydi'odyiiariiics, 34, 283 
Manning, K. V.. .".Id 
Marconi, (!., 157 
Mars, HI, 12 
Marsdon. E., 203 
Mas.s. atomic. 70 

niolcnilar, 7'.) 
Mass-energy relation, HIS 
Mass number, I IS 
Mass rat io, 30 
Mass spectrometer, 131, Mil 

Uainbridge type, 152 

Dempster type, 151 



Index 311 



Matter, composition of. 117 

four states' of. 27(i 
Matter wave, 170, 217, 230 
Maxwell..). C. 83, 87, 88. 103, 166, 

lid. 1114 
Miixwell-iloltzinann distribution, 

s:{. 87, 232 
Maxwell's electromagnetic theory, 

186, Mil. 107 

Maxwell's equations. Mil 
Mean live path, 84 
Mechanics, principles of, 21 
Mendeleev, 1). 1., 228 
Mercury, HI. II 
Mesons, 107 

Metals, conduction in, 232 
Meteorites. HI, 13 

and radio waves, 13 
Miehels, W. ('.. 205 

Micbelson interferometer, lst> 
Michelson-Morlev experiment, ISS 
Slilky Way, 17 
Miliicurie. 242 
Millikan, H. A.. 145 
Mills, M. M-, 38 
Missilery, chronology of, 4 
Model rockets, 37 

manufacturers of, 37 

(Seeaiso Rocket) 
Models of molecules, 8(1 
Modern physics, 77 
Mole, 79 

Molecular mass, 7'.i_ 
Molecular volume. 70 
Molecule, 70. 117 

mode! of. 88 
Momentum, angular, 55 
quantisation of, 209, 220 

in relativity theory, 198 
Moon. HI, 12 
Moslev's (aw, 231 
Motion. Newton's laws of, 21, 43 

uniformly accelerated, 22, 42 
Motors, rocket. 27. 32 



(See alxo Hockot | 

NASA (National Aeronauties and 
Space Administration), 7, 8, 32 

National Association of Rocketry, 
37 

Mangle, J. E., 14, 17. 19 

Neptune, 10, 12 
Neutrino, 240 
Neutron, 118, 274 

detection of, 263 

discovery of, 253 
Newell. H. K.. 14, 17. HI 
Newton, Isaac. (HI 

Newton (unit), 21 

Newton's law of gravitation. II 



Newton's laws of motion, 21, 43 
Node, 51 

Nonionizing radiation, 261 
Nozzle, rocket, 2S, 33 

(See also Hocket) 
Nuclear atom model, 205, 238, 253 
Nuclear landing energy, 200 
Nuclear emulsions, 203 
Nuclear fission, 273 
Nuclear force, 206, 254 
Nuclear propulsion, 31, 71 
Nuclear reactions, energy from, 243. 
25S. 273. 270 

equations for. 25ti 

threshold energy for, 258 
Nuclear reactor, 32, 273, 274 

breeder, 274 
Nuclear testing, 269 
Nuclear wastes, 270 
Nuclei, stable, 255 
Nut ■icon. 1 17, 254 
Nucleus, 1 17, 230 

radius of, 254 

Oersted, B.C., 110 
Orbit, circular, Hi, 53 
electron, 227,230 
for entry o[ atmosphere, 72 
in hydrogen atom. 2111), 222 
satellite, 47 
conic, 66 
elements of, 61 
precession of, 57 
Ordwnv. I 1 '. L. * 
Orear, .1., 63, 296 
Oslwald, W., 281 

Page, I.., S3 

Pair production. 243, 266 
Pairing of aueleons, 255 
Parke, X. <■',., 296 
Pauli exclusion principle, 221 
Performance of liquid propellents, 

20 
Perigee, 51 
argument of, 53 

Period. 51. 54,04 

Periodic table, 228 

Periodicals, io astronautics, 8 

list of, 296 
Permeability, 131 
Permittivity, 1 10 
Phase veloeitv. 100 
Photoelectric effect, 172, 266 
Pholooleotron, 172,242,286 
Photons, 172, 173, 179, ITS, 180,211 

absorption of. 205 
Photosphere, 1 1 
Physical const ants, 304 
Pinch effect, 270 



312 



Index 



Planck, M.. 171 

I'liini-k's constant. 17-'. 173. I7(i 

182 
Planck's law, 170 
Planets, flight between, 68 

life on, 18 

physical data for, 10 

limes to reach, 70 

Plasma, 275 

Plasma propulsion, 35 

Plato, 07 

Poinenre, II., 07, 139, 191 

Pollard, !■:.. 273 

Positron, absorption of. 207 

Positron -electron pair, 243, 205 

Potential, electric, 122 

Potential energy, gravitational, II. 

47 
Potential well, in gravitational Belda, 
47 
for hydrogen atom, 210 
Pouudal. 21 
Poyntnuj'a vector, 166 
Pressure of radial ion, i:t 
Pressure thrust, 24 
Price, 1). J„ 2!iti 

Probability, wave. |8I) 

Product vector, 66 
Project Hover, 32 
Project Sherwood, 34 
Projectile motion, 43 
I'ropellants. performance of, 28 
Propulsion, ion, 34, 3(1 

nuclear, 31 
:, 

solar, 36 
Proton, UK, 127 

Proton-neutron diagram, 243, ->|>(* 
front's hypothesis, 153 

Q- value. 243, 25.S 
Quantization, energy, 208, "242 

momentum, 209, 220 

apace, 224 
Quantum, 172, 173, 170 
Quantum mechanics, 1X4. 2 Hi. 219 
Quantum numbers, 224. 227 
Quantum theory, iu7 

Rad, 268 

liailialioa. atmospheric contamina- 
tion by, 269 
liiologieai effects of, 268 
blnckhody, 108 
cavity, 169 
electromagnetic theory of, 156 

101, 167 
in space, 14 
types of, 20 I 

Radiation holts, 14 



Radiation dose, 267 
Radiation pressure, 13 
Hadiai ion tolerances, 17 
Radioactive equuibn 250 

Radioactive scries, 240 
Radioactivity, decav lav, for, 239 
natural. 238. 249 
series in, 2-10 
lypes of, 239 
units of, 24 I 
Radioisotopes, natural, 251 
Haniu, S., 74 
ISAM) Corporation, 5, 7 
Range, of charged particles, 266 
of rockets. 2! 1. 30. 40, 201 
for atmospheric entry, 73 
Rationalised units, 1 10 
Hay. 93, !!4 

Rayleigh-Jeans Jaw, 17(1 
Reaction principle', 20, 22 
Reaction thrust, 2so 
Reactor, nuclear. 273, 274 

for rocket power. 32 
Reduced mass, 213 
Re-entry of earth's atmosphere 72 
Reference systems, 185 
Relativistic Doppler effect, 114 
Relativity theory, Einstein's, I 111 

ma.vf ami energv in, IVx 
Newton's, 185 
and spaee travel, 201 
I win paradox in, I!I5 
velocity addition in, 103 
Rep, 2tl.s 

Resistance, of free space, 165 

of metals, 147 
Resistivity, 147 
Resnick, *R.. 2116 
Resonance, 1 10 
HiKht-hiiiid rule, 134 
R MS speed. 83 
Roberta, Michael, 1K3 
Hocket, burnout velocity for, 20 

definition of, 20 

flight theory of, 2 1 , 20 

forces on, 29 

ion, 34 

mass ratio, 30 

model, manufacturers of, 37 

motors for. 27. 32 

multiple stage, 31 

oosale, 28, 33 

nuclear, 31 

performance of, 2ii. 36 

plasma, 35 

propulsion of, 21, 22, 26, 32 

range of. 2IJ, 30 

specific impulse of, 25, 27 

staging of, 31 
Rocket guidance, 70 






Rocket trajectories, 60 
Roentgen, w, C, 77 

Roentgen, 267 

Rogers, H. M., 64, 206 

Holler, I). H. I)., 295 

Hollett, J. S., 304 

Root-in can -square speed (rtns), 83 

Rouse, [..,!.. 296 

Rowland, H. A., 2t)(i 

Russell. Hortrand, 183 

Rutherford atom model, 205, 238, 

2S3 
Ryilbcrg constant, 207 

Sarnoff. !>.. 149 
Satellite orbits, 47 
Satellites, earth. 3 

energy of, 55 

escape speed for, 31 

reasons for, 0, 50 
Saturn, 10, 12 
.Savage, J. N„ 20 
Scattering of alpha particles, 204 
Schoolcy, J. S., 48 
Sehrodingcr, [■',., 216 
Sehrodiiigcr wave equation, 222, 

220, 245, 203 
Scintillation counter, 262 
Screening, electron. 220 
Scebeek, T., 277 
Sciferl, B. §., 8,88, 58 
Seismic wave, 103 
Semiconductor, 237, 277 
Series, radioactive, 240 
Shaw, J. H., 10 
Shells, electron. 227, 230 
Shock wave, 102 
Simultaneity, 103 
Singer, ('Italics, 13N 
Slide rule, use of, 298 
Slug, 21 

Societies, professional. 296 
Solar const ii at. 1 1 
Solar propulsion, 30 
Solar sail, 36 
Solar system. 10. 18 

Soli. I stale, theory ..I'. 231 

Sonic boom, 102 
Sound wave. 100 

speed of, 101 

0e» oho Wave) 
Space, environment of, 

gravitational fields in. 40, 47 

radiation in, 14 

vehicles in, 4 
Space exploration, reasons for, li. 
71 

Spaee quantization, 224 

Space research, 74 

Space travel, and relativity, 201 



Index 313 

Space vehicles, chronology of, 4 
Spec i lie heat, 87 

of metals, 147 
Specific impulse. 25. 27, 32, 33 
Spectrum, I0S, 2<l(» 

of hydrogen. 207 

x-ray. 230 
Speed, burnout, 201 

of light. Hi.!. ISO. I!I2 

of molecules. S3, SO 

rms, 83 
Spencer, II., 67 
Spin, 225. 249 
Sputniks, 5, 50, 05 
Stable nuclei. 255 
Stages, rocket, 31 

(ate also Rocket | 
Standing wave, I (19, 1 1 1 
Star distances, 18 
State, equation of, SI 
Stefan-Boltsmann law, 170 
Stern, ()., 86 

Storn-dei huh experiment. 224 
SlMwell. 0. EL III 
Stake's law, 145 
Stouer. R. <:.. 153 
Stoney, G. J., 1 44 
Strassmann, F„ 273 
Strughold, II., 7 
Suninierfeld, M., 38 
Sun. 10 

Superposition principle, 104. 109 
Surge. 9S 

Sutton. G. P., 26 

Symbols, mathematical, 208 

Thermion ie converter, 281 
Thermocouple, 278 
Thermoelectricity. 277 

power from, 279 
Thomas, B, It., !K3 
Thompson, II. !.., 30(1 
Thomson, J, J., 77, 131, 130, 149, 

203. 288 
Tli res hold energy, 25s 
Thrust, decrease with alt it talc, 24 

momentum, 24 

pressure. 24 

rocket, 24, 289 
Time dilatation. 114. 195. 201 
Time to reach planets. 711 
Tiros satellite. 58 
Total energy of B parliele, 2(10 
Transformations, Lorentz, 192 
Transurauic elements, 273 
Trigonometric functions, 301 
Tunnel effect . 245 
Twin paradox, 105 
Tyndall, J., 116 



314 



Index 



I'liceifainty principle. ISI 
l.'nifoniily accelerated motion, 22, 

42 
("nils, electrostatic, 119 

tuks, 119, 120 

for Newton's second law, 21 

rationalized mks, I li) 
Utmma (film), 18 
Drams, 10, 12 

\'-2 rocket, 24, 30 
\an Allen belts, 14, 16 
Vector cross product, 50 
Vector product, !2,s 
Velocities, iiddition of, 193 
Velocity, burnout, 29 
of escape, 45 
group, 1 1 in 
of molecules, H'A, so 
phase, Hit) 
rms, 83 
Venus, JO, II 
Volt, 122 
Von Goethe, J. W., 115 

Wave, acoustic, 101, 108 
at a boundary, 04, 110, 245 
damped, 106 
in different mediums, 94 
elastic, 93 

electromagnetic, 93, 103, 101 
energy of, 105 
group velocity in, 100 
intensity of, 105 
in a liquid, 07 
longitudinal, 92, 103 
panicle motion in. 99 
phase velocity in, 100 
saw-tool li, 90 
seismic, 103 
shock, 102 
sine, 96 



Wave, sound, 100 

speed of, 104 

square, 96 

standing 109, III 

in u string, 95, 1 11 

surge, !IS 

transverse, 92, 103 

traveling, 95 

x-ray, 17.5 
Wave aquation, 92, no 

Schroilinger. 293 
Wave forms, 95, 90, 104 
Wave front, 93. :it 
Wave mechanics, 219, 245, 293 
\\.ni-|i;iniile duality, 179, 217 
W ave speed, !(ll 
Waves, interference of, Km 

soperposi'i f. KM, 109 

Weber, R. I.„ 7, 153,290 
Weiicr, 129 
Weightlessness, \i\ 
Wells. II. Q 00 
Whipple, V. L_ 3 
White, M. \V., 290 
Whitehead, A. \., 50 
Whitney, W. \i., 20 
Whole number rule-. 153 
Wiedcnianu-Frani! rule, 233 
\\ nil's law. 171 
Wilson, V. C, 281 
Wilson cloud chamber, 262 

X-ray spectrum, characteristic, 230 

continuous, 230 
X rays, 175 

frequency limit of, 176 

scattering of, J 76 

ffin, II., 19 
Zodiac, 9 
Zwicky, h\, 08 



PHYSICAL DATA, EARTH 

Mean diameter 12,742.46 km 

Angular velocity 72.9 X 10~ 6 radian/sec 

Mass 5.975X10" kg 

Mean density 5,517 gm/cm 3 

Normal gravity (p = geodetic latitude) 

g m 9.78049(1 + 5.22884 X 10- 3 sin 3 ^ - 5.9 X lO" 6 sin 2 2<p) 

m/sec 2 
Standard atmosphere p = 1.013 X 10 B newton/m 2 



PHYSICAL CONSTANTS 

Na = Avogadro's number = 6.0249 X 10 M molecules/kmole 

R = Gas constant per mole = 8317 joules/ (kmole)(°K) 

k - Boltzmann constant m 1.3804 X 1Q~ 23 jou1e/°K = 8.617 X 

10- 5 ev/°K 
y Q = Standard volume of a perfect gas = 22.420 m 3 afm/kmole 
c - speed of light - 2.9979 X 10 s m/sec 
e m Electronic charge m 1.6021 X 10~ 19 coul 
h » Planck's constant = 6.6252 X 10~ 34 joule-sec 
F = Faraday constant « 9.652 X 10 T coul/kmole 

Energy conversions: 1 electron volt = 1.6021 X 10- 19 joule 
1 atomic mass unit = 931.14 Mev 

m c = Rest mass of electron - 9.1083 X 10~ 3I kg = 0.51098 Mev 
m p = Rest mass of proton = 1.6724 X 10"" kg - 938.21 Mev 
Mb - Rest mass of neutron m 1.6747 X 10-" kg - 939.51 Mev 
mjm s = 1836.12 

e = — c 1 = Permittivity of free space = 8.8542 X 10" 12 

farad/m 
^o = Permeability of free space = 4tt X 10" 7 henry/m 
Zo = (/*oAo)i = Impedance of free space = 376.73 ohm 
G = Universal gravitational constant = 6.67 X 10" n 

newton-mVkg 2 



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